STATISTICS (M)

1) SOME BASIC TERMS AND CONCEPTS 

  a) An Experiment:

An action or operation resulting in two or more outcomes is called an experiment.

 

   b) Sample Space:

The set of all possible possible all possible possible outcomes of an experiment is experiment is called the sample space, denoted by S. An element of S is called sample point.

  c) Event :

Any subset of sample space is an event.

  d) Simple Event:

An event is called simple event if it is a singleton subset of subset of sample space S.

  e) Compound Events:

It is the joint occurrence of two or more simple events.

  f) Equally Likely Events:

A number of simple events are said to be equally likely if there is no reason for one event to occur to occur occur in preference to any other event.

  g) Exhaustive Events:

All the possible outcomes taken together in which an experiment can result as said to be exhaustive or disjoint.

  h) Mutually Exclusive or Disjoint Events:

If two events cannot occur simultaneously, then they are mutually exclusive. If A and B are mutually exclusive then A∩B=∅

  i) Complement of an Event:

The complement of an event A, denoted by A' , A'' or Aᶜ is the set of all sample points of the space other than the sample points in A

2) MATHEMATICAL DEFINITION OF PROBABILITY

      Let the outcomes of an experiment consists of n-exhaustive mutually exclusive and equally likely cases. Then the sample spaces S has n sample points. If an event A consists of m sample points, (0 ≤ m ≤ n), then the probability of event A, denoted by P(A) is defined to be m/n i.e., P(A) = m/n.

  let S= a₁, a₂, .... aₙ be the sample space.

     a) P(S)= n/n= corresponding to the certain event.

     b) P(∅)= 0/n =0 corresponding to the null event ∅ or impossible event.

    c) If Aᵢ = {aᵢ}, i = 1, .... n then Aᵢ is the event corresponding to a single sample point aᵢ , then P(A₁) = 1/n

   d) 0 ≤ P(A) ≤ 1.

3) ODDS AGAINST AND ODDS IN FAVOUR OF AN EVENT:

   

      Let there be be e m + n equally likely, mutually exclusive and exhaustive cases out of which an event A can occur in m cases and does not occur in n cases. Then by definition of probability of occurrences = m/(m+n)

  The probability of non-occurance = n/(m+n)

  So, P(A): P(A')= m: n

 Thus, the odds in favour of occurrences of the event A are defined by m:n i.e., P(A) : P(A') ; and the odds against the occurrence of the event A are defined by n: m i.e., P(A') : P(A)

4) ADDITION THEOREM:

    a) A and B are any event in S, then P(AUB) = P(A)+ P(B) - P(A∩B)

  Since the probability of an event is a non negative number, it follows that 

P(A UB) ≤ P(A) + P(B)

For three events A, B and C in we have P(AUBUC)= P(A)+ P(B)+ P(C) - P(A∩B) - P(B∩C) - P(C∩A) + P(A∩B∩C).

General Form Of Addition Theorem -->

  For n events A₁, A₂, A₃.... Aₙ in S, we have P(A₁UA₂)U A U A ... Aₙ)

= ⁿᵣ₌₁∑ P(Aᵢ)= ᵣ₌₁ ∑P(Aᵢ ∩Aᵢ) + ᵢ₌ⱼ₌ₖ∑ P(Aᵢ∩Aⱼ ∩ Aₖ) ..... + (-1)ⁿ⁻¹ P(A₁∩ A₂∩A ₃ .......∩ Aₙ)

   b) If A and B are mutually exclusive, then P(A∩ B)= 0 so that P(AUB) = P(A)+ P(B)

5) MULTIPLICATION THEOREM:

    Independent event:

So, if A and B are two independent events that happening of B will have no effect on A.

      Difference Between Independent & Mutually Exclusive Event:

    (i) Mutually exclusiveness is used when events are taken from same experiment & Independence when events one takes from different experiment.

   (ii) Independent events are represented by word "and" but mutually exclusive events are represented by word "OR" 

    (a) When Events are Independent: 

P(A/B) = P(A) and P(B/A)= P(B), then

P(A ∩ B)= P(A) . P(B). OR

P(AB) = P(A) . P(B)

     

      (b) When Events Are Not Independent-->

    The probability of simultaneous happening of two events A and B is equal to the probability of A multiplied by the conditional probability of B with respect to A (or probability of B multiplied by the conditional probability of A with respect to B) i.e.,

 P(A ∩ B)= P(A) . P(B/A). OR

 P(B) . P(A/B).    

          OR

P(AB)= P(A).P(B/A). OR

P(B) . P(A/B)

   c) Probability Of At Least One Of The n Independent Events-->

             IF P₁ .P₂.P₃......Pₙ are the probabilities of n independent events A₁. A₂.A₃....Aₙ then the probability of happening of at least one of these event is

     1 - [(p₁)(1- p₂)....(1 - pₙ)]

P[A₁ +A₂+A₃+ .....+ Aₙ) = 1 - P(A'₁) . P( A'₂). P(A'₃)........P(A'ₙ) 

6) CONDITIONAL PROBABILITY: 

     Conditional Probability:

 If A and B are any events in S then the conditional probability of B relative to A is given by

  P(B/A)= P(B∩A)/P(A) , If P(A)≠0

7) BAYE'S THEOREM OR INVERSE PROBABILITY:

  Let A₁, A₂, ........Aₙ be n mutually exclusive and exhaustive events of the the of the sample space S and A is event which can occur with any of the events then

 P(Aᵢ/A) = {P(Aᵢ) P(A/Aᵢ)}/ {ⁿᵣ₌₁∑P(A₁) P(A/A₁)}

7) BINOMIAL DISTRIBUTION FOR REPEATED TRIALS:::

 Binomial Experiment:

 Any experiment which has only two outcomes is known as binomial experiment.

Outcomes of such an experiment are known as success and failure probability of success is denoted by p and probability of failure by q. p+ q = 1

If binomial experiment is repeated n times, then (p+q)ⁿ =ⁿC₀qⁿ + ⁿC₁pqⁿ⁻¹ + ⁿC₂p²qⁿ⁻² + ....... + ⁿCᵣpʳqⁿ⁻¹ + ....... + ⁿCₙ pⁿ = 1

     a) Probability of exactly r successes in n trials =ⁿCᵣpʳqⁿ⁻¹ 

    b) Probability of at most r successes in n trials = ʳₖ₌₀∑ⁿCₖ pᵏ qⁿ⁻ᵏ 

   c) Probability of at least r success in n trials= ʳₖ₌₀∑ⁿCᵣpᵏqⁿ⁻ᵏ

 

   d) Probability of having 1ˢᵗ success at the rᵗʰ trials= p qʳ⁻¹

         The mean the variance and the standard deviation of binomial distribution are np, npq, √(npq) .

9) SOME IMPORTANT RESULTS:

     a) Let A and B be two events, then 

  (i) P(A)+ P(A') = 1

 (ii) P(A+B)= 1 - P(A'B')

 (iii) P(A/B)= P(AB)/P(B) 

(iv) P(A+B)=P(AB)+P(A'B)+P(AB')

(v) A⊂B⇒P(A) ≤ P(B)

(vi) P(A' B)= P(B) - P(AB)

(vii) P(AB)≤ P(A) P(B) ≤ P(A+B) ≤ P(A)+ P(B)

viii) P(AB)= P(A)+ P(B) - P(A+B)

ix) P(exactly one event)= P(A B') + P(A' B)

  = P(A)+ P(B) - 2P(AB)

  = P(A+B) - P(AB)

x) P(neither A nor B) = P(A' B')

             = 1 - P(A+B)

xi) P(A'+ B') = 1 - P(AB)

   b) Number of exhaustive cases of tossing n coins simultaneously (or of tossing a coin n times)= 2ⁿ

   c) Number of exhaustive cases of throwing n dice simultaneously (or throwing one dies dies n times)= 6ⁿ

   d) Playing Cards:

      i) Total cards: 52(26 red, 26 black)

      ii) Four suits: Heart, Diamond, Spade, Club-- 13 cards each.

     iii) Court cards: 12(4 Kings, 4 Queens, 4 jacks)

     iv) Honour Cards:16( 4 aces, 4 Kings, 4 Queens, 4 Jacks)

   e) Probability Regarding n Letters and their envelopes:

          if n letters corresponding to n envelopes placed in the envelopes at random, then

     i) Probability that all letters are in right envelopes= 1/n!

     ii) Probability that all letters are not in right envelopes= 1- 1/n!

     iii) Probability that no letters is in right envelopes = 1/2! - 1/3! + 1/4! - ......+ (-1)ⁿ 1)n!

     iv) Probability that exactly r letters are in right envelopes

= 1/r![1/2! - 1/3! + 1/4! - ....+(-1)ⁿ⁻ʳ 1/(n -r)!]

 

Exercise -A

1) An unbiased dice is thrown. What is the probability of getting:

A) an even number.                    1/2

B) a multiple of 3.                       1/3

C) an even number or a multiple of 3.                                          2/3

D) an even number and a multiple of 3.                                         1/6

E) a number 3 or 4.                      1/3

F) an odd number.                       1/2

G) A number less than 5.             2/3

H) A number greater than 3.        1/2

I) A number between 3 and 6.     1/3

2) Two unbiased coins are tossed simultaneously. Find the probability of getting:

A) Two heads.                        1/4

B) one head.                           1/2

C) one tail.                              1/2

D) at least one head.             3/4

E) at most one head.             3/4

F) No head.                             1/4

3) Three unbiased coins are tossed together. Find the probability of getting:

A) all heads.                                1/8

B) two heads .                             3/8

C) one head.                                3/8

D) at least two heads.                1/2

4) Two dice are thrown simultaneously. Find the probability of getting:

A) an even number as the sum. 1/2

B) the sum as the prime number. 5/12

C) a total atleast 10.                     1/6

D) a doublet of even number.    1/12

E) A multiple of 2 on one dice and a multiple of 3 on the other.       11/36

F) same number on the both dice. 1/6

G) a multiple of 3 as the sum.    1/3

5) Find the probability that a leap year selected random, will contain 53 Sundays.                              2/7

6) What is the probability that a number selected from the numbers 1,2,3,..... 25 is a prime number, when each of the given numbers is equally likely to be selected.     9/25

7) Tickets numbered from 1 to 20 are mixed up together and then a ticket is drawn at random. What is the probability that the ticket has a number which is a multiple of 3 or 7 ?              2/5

8) A pack of playing cards consists of 52 cards, each of the 52 cards being equally likely to be drawn. Find the probability that the card drawn is:

A) An ace.                                 1/13

B) red.                                         1/2

C) either red or King.                7/13

D) red and a king.                      2/26

E) a face card.                            4/13 

F) a red face card.                     2/13

G) 2 of spade.                             1/12

H) 10 of a black suit.                 1/26

9) The king queen and jack of clubs are removes from a deck of 52 playing cards and the well shuffled. One card is selected from the remaining cards. Find the probability of getting:

A) A heart.                                13/49

B) a king.                                    3/49

C) a club.                                   10/49

D) the 10 of Hearts.                   1/49

10) A bag contains 3 red and 2 blue marbles. A marble is drawn at random of drawing a blue marble.         2/5

11) A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of a red ball, Find the number of blue balls in the bag.                              10

12) A bag contains 12 balls out of which x are white.

A) If one ball is drawn at random, what is the probability that it will be a white ball.                                x/12

B) if 6 more white balls are put in the bag, the probability of drawing a white ball will be double than that A. Find x.   3                                   

13) It is know that a box of 600 electric bulbs contains 12 defective bulbs. One bulb is taken out at a random from this box. What is the probability that it is a non defective bulb ?                     0.98

14) 17 cards number 1,2,3,....17 are put in a box and mixed thoroughly. One person draws a card from the box. Find the probability the number on the card is :

A) odd.                                         9/17

B) a prime.                                   7/17

C) divisible by 3.                          5/17

D) divisible by 3 and 2 both.      2/17

15) Cards marked with the numbers 2 to 101 are placed in a box and mixed thoroughly. One card is drawn from this box. Find the probability that the number on the card is:

A) an even number.                    1/2

B) a number less than 14.       3/25

C) a number which is a perfect square.                                      9/100

D) A prime number less than 20.    2/25

16) 1000 tickets of a lottery were sold and there are 5 prizes on these tickets. If Saket has purchased one lottery ticket, what is the probability of winning a prize ?                0.005

17) A child has a block in the shape of a cube with one letter written on each face as shown

 A B C D E A

The cube is thrown once. What is the probability getting

A) A.                                            1/3

B) D.                                            1/6

18) A bag contains 5 red balls, 8 white balls, 4 green balls and 7 black balls. If one ball is drawn at random, find the probability that it is:

A) black.                                     7/24

B) red.                                         5/24

C) not Green.                              5/6

19) A die is thrown. Find the probability of getting:

A) a prime number.                    1/2

B) two or four.                            1/3

C) a multiple of 2 or 3.              2/3

20) In a simultaneous throw of a pair of dice, find the probability of getting:

A) 8 as the sum.                         5/36

B) a doublet.                                 1/6

C) a doublet of prime numbers.  1/12

D) a doublet of odd numbers.  1/12

E) a sum greater than 9.              1/6

F) an even number on first.         1/2

G) an even number on one and a multiple of 3 on the other.       11/36

H) neither 9 nor 11 as the sum of the numbers on the faces.         5/6

I) a sum less than 6.                  5/18

J) a sum less than 7.                 5/12

K) a sum more than 7.               5/12

21) Three coins are tossed together. Find the probability of getting:

A) exactly two heads.                   3/8

B) at least two heads.                  1/2

C) at least one head and one tail.  3/4

22) what is the probability that an ordinary year has 53 Sundays ? 1/7

23) what is the probability that a leap year has 53 Sundays and 53 Mondays.                       1/7

24) A and B throw a pair of dice. If A throws 9, find B's chance of throwing a higher number.         1/6

25) Two unbiased dice are thrown. Find the probably that the total of the numbers on the dice is greater than 10.                   1/12

26) A card is drawn at random from a pack of 52 cards. Find the probably that the card is drawn is:

A) a black King.                         1/26

B) either a black card or a king.    7/13

C) black and a king.                    1/26

D) a Jack, Queen or a King.       3/13

E) neither a heart nor a king.     9/13

F) spade or an ace.                    9/13

G) neither an ace nor or a king.    11/13

27) In a lottery 50 tickets numbered 1 to 50, one ticket is drawn. Find the probability that the drawn ticket bears a prime number.             3/10

28) An urn contains 10 red and 8 white balls. One ball is drawn at random. Find the probability that the ball drawn is white.              4/9

29) A bag contains 3 red balls, 5 black balls and 4 white balls. A ball is drawn at random from the bag. What is the probably the ball drawn is:

A) white ?                                   1/3

B) red ?                                       1/4

C) black ?                                   5/12

D) not red ?                                 3/4

30) What is the probability that a number selected from the numbers 1, 2, 3,.....15 is a multiple of 4 ?   1/5

31) A bag contains 6 red, 8 black and 4 white balls. A ball is drawn at random. What is the probability that ball drawn is not black ?            5/9

32) A bag contains 5 white and 7 red balls. One ball is drawn at random. What is the probability that ball drawn is white.                   5/12

33) Tickets number from 1 to 20 are mixed up and a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 3 or 7 ?     2/5

34) In a lottery there are 10 prizes and 25 blanks. What is the probability of getting a prize.      2/7

35) if the probability of winning a game is 0.3, what is the probability of loosing it.                                 0.7

36) A bag contains 5 black, 7 red and 3 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is:

A) red.                                      7/15

B) Black or white.                   8/15

C) not black.                             2/3

37) A black contains 4 red, 5 black and 6 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is

A) white.                                     2/5

B) red.                                        4/15

C) not black.                               2/3

D) red or white.                           2/3

38) Fill in the blanks:

a) Probability of a sure event is--

b) probability of an impossible event is ____

c) The probability of an event (other than sure and impossible event) lies between ___

d) Every elementary event associated to a random experiment has____ probability.

Answer: 1, 0, 0 and 1 , equal

Exercise -B

1) Find the probability of getting a head in a toss of an unbiased coin.    1/2

2) In a simultaneous toss of two coins, find the probability of getting:

A) 2 heads.                                     1/4

B) exactly one head.                     1/2

C) exactly 2 tails.                           1/4

D) exactly one tail.                        1/2

E) no tails.                                      1/4

3) Three coins are tossed once. Find the probability of getting:

A) all heads.                                   1/8

B) atleast two heads.                    1/2

C) atmost two heads.                  7/8

D) no heads.                                  1/8

E) exactly one tail.                        3/8

F) exactly two tails.                       3/8

G) a head on first coin.    

H) atleast one head and one tail.    3/4

4) A die is thrown. Find the probability of getting:

A) an even number.                      1/2

B) a prime number.                      1/2

C) a number greater than or equal to 3.                             2/3

D) a number less than or equal to 4.        2/3

E) a number more than 6.              0

F) a number less than or equal to 6.        1

G) 2 or 4.                                        1/3

H) A multiple of 2 or three.         2/3       

5). Two dice are thrown simultaneously. Find the probability of getting:

A) an even number as the sum.  1/2

B) the sum as a prime number. 5/12

C) a total of atleast 10.                1/6

D) a doublet of even number.    1/12

E) a multiple of 2 on one dice and a multiple of 3 on the other dice.    11/36

F) same number on both dice.   1/6

G) a multiple of 3 as the sum.    1/3

H) neither a doublet nor a total of 8 will appear.                     13/18

I) the sum of the numbers obtained on the two dice is neither a multiple of 2 nor a multiple of 3.             1/3

6) Three dice are thrown together. Find the probability of getting:

A) a total of atleast 6.          103/108

B) a total of 17 or 18.                1/54

7) What is the probability that a number selected from the numbers 1, 2, 3,...., 25, is prime number, when each of the given numbers is equally likely to be selected?    9/25

8) Tickets numbered from 1 to 20 are mixed up together and then a ticket is drawn at random, what is the probability that the ticket has a number which is a multiple of 3 or 7.                   2/5

9) A coin is tossed. If head comes up, a die is thrown but if tail comes up, the coin is tossed again. Find the probability of getting:

A) two tails.                                   1/8

B) head and number 6.                1/8

C) head and an even number.     3/8

10) A letter is chosen at random from the word ASSASSINATION. Find the probability that letter is

A) a vowel.                                   6/13

B) a consonant.                          7/13

11) In a lottery, a person choses six different natural numbers at random from 1 to 20. And if these six numbers match with the six numbers already fixed by the lottery committee, he wins prize. What is the probability of winning the prize in the game?                         1/38760

12) On her vacations Ram visits four cities A, B, C , D in a random order. What is the probability that he visits.

A) A before B.                                1/2

B) A before B and B before C.     1/6

C) A first and B last.                   1/12

D) A either first or second.          1/2

E) A just before B.                        1/4

13) A die has two faces each with number '1' three faces each with number '2' and one face with number '3'. If die is rolled once determine:

A) P(2).                                         1/2

B) P(1 or 3).                                 1/2

C) P(not 3).                                  5/6

14) If 4-digit numbers greater than or equal to 5000 are randomly formed from the digits 0, 1, 3, 4 and 7, what is the probability of forming number divisibile by 5 when

A) the digits may be repeated.    2/5

B) the repetation of digits is not allowed.                     3/8

15) One card is drawn from a pack of 52 playing cards, each of the 52 cards being equally likely to be drawn. Find the probability that the card drawn is:

A) an ace.                                  1/13

B) red .                                         1/2

C) either red or king.                 7/13

D) red and a king.                     1/26

16) An urn contains 9 red, 7 white and 4 black balls. If two balls are drawn at random, find the probability that:

A) both the balls are red.         18/95

B) one ball is white.                91/190

C) the ball are of the same colour.    63/190

D) one is white and other red.    63/190

17) A bag contains 6 red, 4 white and 8 blue balls. If three balls are drawn at random, find the probability that

A) one is red, one is white and one is blue.                                       4/17

B) one is red and two are white.   3/68

C) two are blue and one is red. 7/34

D) one is red.                            33/68

18) A bag contains 4 red, 7 white and 5 black balls. If two balls are drawn at random, find the probability that 

A) both the balls are white.       7/40

B) one is black and the other red.    1/6    

C) both the balls are of the same colour.                                    37/120

19) A box contains 10 red marbles, 20 blue marbles and 30 green marbles. 5 marbles are drawn from the box, what is the probability that

A) all will be blue.           20C5/60C5

B) at least one will green?               1 - 30C5/69C5

20) A box contains 8 red, 3 white and 9 blue balls. 3 balls are drawn at random, what is the probability that

A) all the three balls are blue balls.     7/95

B) all the balls are of different colours.                                  18/95

21) A box contains 5 red marbles, 6 white marbles and 7 black marbles. 2 marbles are drawn from the box, what is the probability that both the balls are red or both are black.      31/153

22) In a lottery 10000 tickets are sold and ten equal prizes are awarded. What is the probability of not getting a prize if you buy

A) 1 ticket.                         999/1000

B) two tickets.      9990C2/10000C2

C) 10 tickets.    9990C10/10000C10

23) The number of lock of a suitcase has 4 wheels, each labelled with ten digits i.e., from 0 to 9. The lock opens with a sequence of four digits with no repeats. What is the probability of a person getting the right sequence to open the suitcase.          1/5040

24) Out of 100 students, two sections of 40 and 60 students are formed. If you and your friends are among the 100 students, what is the probability that

A) you both enter the same section.         17/33

B) you both enter the different sections?                                 16/33

25) Four cards are drawn at random from a pack of 52 playing cards. Find the probability of getting:

A) all the four cards of the same suit.                                   198/20825

B) all the four cards of the same number.                            13/270725

C) one card from each suit.    2197/20825

D) two red cards and two black cards.              (26C2 x 26C2)/52C4.

E) all cards of the same colour.    2.26C4/52C4.

F) all face cards.             12C4/52C4

26) In a lottery of 50 tickets numbered 1 to 50, two tickets are drawn simultaneously. Find the probability that:

A) both the tickets drawn have prime numbers.                      21/245

B) none of the tickets drawn has prime number.                           17/35

C) one ticket has prime number. 3/7

27) In a lottery, a person chooses six different numbers at random from 1 to 20, and if these six numbers match with six numbers already fixed by the lottery committee, he wins the prize. What is the probability of winning the prize in the game?             1/38760

28) A word consists of 9 letters, 5 consonants and 4 vowels. Three letters are chosen at random. What is the probability that more than one vowel be selected.         17/42

29) Four persons are to be selected at random from a group of 3 men, 2 women and 4 children. Find the probability of selecting:

A) 1 man, 1 woman, 2 children.  2/7

B) exactly 2 children.                10/21

C) 2 women.                                   1/6

30) A box contains 10 bulbs, of which just three are defective. If a random sample of five bulbs is drawn, find the probabilities that the sample contains:

A) exactly one defective.          5/12

B) exactly two defective.          5/12

C) no defective bulbs.              1/12

31) A box contains 100 bulbs, 20 of which are defective. 10 bulbs are selected for inspection. Find the probability that:

A) all 10 are defective.   

B) all 10 are good.

C) atleast one is defective.

D) none is defective.   

32) A bag contains tickets numbered 1 to 30. Three tickets are drawn at random from the bag. What is the probability that the maximum number of the selected tickets exceeds 25.               88/203

33) Twelve balls are distributed among three boxes, find the probability that the first box contains three balls. (12C3 x 2⁹)/3¹²

34) Five marbles are drawn from a bag which contains 7 blue and 4 black marbles. Find the probability that:

A) all will be blue.                       1/22

B) 3 will be blue and 2 black.    5/11

35) Find the probability that when a hand of 7 cards is dealt from a well shuffled deck of 52 cards. It contains

A) all 4 kings.                        1/7735

B) exactly 3 kings.                9/1547

C) atleast 3 kings.              46/7735

36) In a single throw a three dice, determine the probability of getting:

A) a total of 5. 1/36

B) a total of atmost 5.             5/108

C) a total of atleast 5.              53/54

37) Three dice are thrown simultaneously. Find the probability that:

A) all of them show the same face.         1/36

B) all show distinct faces.           5/9

C) two of them show the same face.              5/12

38) What is the probability that in a group of

A) 2 people both will have the same birthday?                                1/365

B) 3 people, atleast two will have the same birthday.                    (364x 363)/365²

39) The letters of the word SOCIETY are placed at random in a row. What is the probability that three vowels come together.      1/7

40) The letters of the word SOCIAL are placed at random in a row. What is the probability that the vowels come together.                            1/5

41) The letters of the word CLIFTON are placed at random in a row. What is the probability that the vowels come together.                2/7

42) The letters of the word FORTUNATES are placed at random in a row. What is the probability that the two T come together. 

43) The letters of the word UNIVERSITY are placed at random in a row. What is the probability that two I do not come together.        4/5

44) Find the probability that in a random arrangement of the letters of the word UNIVERSITY the two I's come together.                            1/5

45) A five digits number is formed by the digits 1,2,3,4,5 without repetition. Find the probability that the number is divisible by 4.        1/5

46) Out of 9 outstanding students in a college, there are 4 boys and 5 girls. A team of four students is to be selected for a quiz programme. Find the probability that two are boys and two are girls.             10/21

47) There are 4 letters and 4 addressed envelopes. Find the probability that all the letters are not dispatched in right envelopes.      23/24

48) The odds in favour of an event are 3:5. Find the probability of occurrence of this event.          3/8

49) The odds in favour of an event are 2:3. Find the probability of occurrence of this event.          2/5

50) The odds in against of an event are 7:9. Find the probability of non-occurrence of this event.          7/16

51) A card is drawn from an ordinary pack of 52 cards and a gambler bets that, it is a spade or an ace. What are the odds against his winning this bet.                   9:4

52) Two dice are thrown. Find the odds in favour of getting the sum

A) 4.                                           1:11

B) 5.                                            1:8

C) What are the odds against getting the sum 6?                   31:5

53) What are the odds in favour of getting a spade if the card drawn from a pack of cards ? What are the odds in favour of getting a king ?         1:3, 1:12

54) A fair coin with 1 marked on one face and on the other and a fair die are both tossed, find the probability that the sum of numbers that turns up is 

A) 3.                                             1/12

B) 12.                                           1/12

55) In a relay race there are five teams A, B, C, D and E.

A) what is the probability that A, B and C finish first, second and third respectively.                              1/60

B) what is the probability that A, B and C are first three to finish (in order).                    1/19

56) In shuffling a pack of 52 playing cards, four are accidently dropped; find the probability that the missing cards should be one from each suit.            2197/20825

57) Five cards are drawn from a pack of 52 cards. What is the probability that these 5 will contain

A) just one ace.          3243/10829

B) atleast one ace.       

58) If a letter is chosen at random from the English alphabet, find the probability that the letters.

A) a vowel.                                 5/26

B) a constant.                          21/26  

59) A class consists of 10 boys and 8 girls. Three students are selected at random. What is the probability that the selected group has

A) all boys.                                 5/34

B) all girls.                                7/102

C) 1 boy and 2 girls.              35/102

D) atleast one girl.                   29/34

E) at most one girl.                  10/17

Exercise - C

Formula:

* (Addition theorem for two events) If A and B are two events associated with a random experiment, then 

1) P(A∪B) = P(A) + P(B) - P(A∩B)

* If A and B are mutually exclusive events, then

P(A∩B) = 0

So, P(A UB) = P(A) + P(B)

This is the addition theorem for mutually exclusive events.

2) (Addition Theorem for three events) If A, B, C are three events associated with a random experiment, then

P(A∪B∪C) = P(A) + P(B) + P(C) - P(A∩B) - P(B∩C) - P(A∩C) + P(A∩B∩C).

* If A, B, C are mutually exclusive events, then

 P(A∩B)= P(B∩C) = P(A∩C) = P(A∩B∩C) = 0.

P(A U BU C)= PA) + P(B)+ P(C).

This is the addition theorem for three mutually exclusive events.

3) i) P(A'∩B)= P(B) + P(A∩B) 

ii) P(A∩B') =P(A) - P(A∩B)

iii) P(A∩B') U P(A'∩B) = P(A) + P(B) - 2P(A∩B) 

* P(A'∩B) is known as the probability of occurrence of B only.

* P(A∩B') is known as the probability of occurrence of A only.

* P(A∩B')U P(A'∩B) is known as the probability of occurrence of exactly one of two events A and B.

* If A and B are two events associated to a random experiment such that A ⊂ B, then A' ∩ B ≠ ∅

4) For any two events A and B

P(A ∩ B) ≤ P(A) ≤ P(A U B) ≤ P(A) + P(B).

5) P(A) + P(B) - P(A UB)) = P(A U B) - P(A ∩B).

6) P(A' ∩ B') = 1 - P(A U B)

+++++++++++++++++++++++++++

1) Given P(A)= 3/5 and P(B) = 1/5, Find P(A or B), if A and B are mutually exclusive events.          4/5

2) A and B are two mutually exclusive events of an experiment. If P(not A)= 0.65, P(A UB)= 0.65 and P(B)= p, find the value of p.        0.30

3) Given P(A)= 1/4 and P(B) = 2/5, P(A U B) = 1/2, find 

A) P(A∩ B).                                 3/20

B) P(A∩ B').                                1/10

if A and B are mutually exclusive events.     

4) If E and F are two events such that P(E)= 1/4, P(F)= 1/2 and P(E and F)= 1/8, find

A) P(E or F).

B) P(not E and not F)

5) If A and B are two events associated with a random experiment such that P(A)= 0.3, P(B) = 0.4, P(A U B) =0.5, find 

A) P(A∩ B).                                   0.2

6) If A and B are two events associated with a random experiment such that P(A)= 0.5, P(B) = 0.3,P(A∩ B)=0.2, find 

A) P(A U B).                                  0.6

7) If A and B are two events associated with a random experiment such that P(AU B)= 0.8, P(A ∩B) = 0.3, P(A')= 0.5, 

find P(B).                                       0.6

8) Given P(A)= 1/2 and P(B) = 1/3, Find P(A or B), if A and B are mutually exclusive events.          5/6

9) Given P(A)= 0.4 and P(B) = 0.5, if A and B are mutually exclusive events associated with a random experiment. Then find

A) P(AU B).                                    0.9

B) P(A' ∩ B').                                 0.1

C) P(A' ∩ B).                                 0.5

D) P(A ∩ B').                                 0.4

10) A and B are two events such that Given P(A)= 0.54 and P(B) = 0.69, P(A ∩ B) = 0.35. find

A) P(AU B).                                0.88

B) P(A' ∩ B').                              0.12

C) P(A ∩ B').                               0.19

D) P(B ∩ A').                                0.34

11) Fill in the blanks:

   P(A) P(B) P(A ∩ B) P(AU B)

A) 1/3 1/5 1/15 ____

B) 0.35 ___ 0.25 0.6

C) 0.5 0.35 ___ 0.7

12) Check whether the following probabilities P(A) and P(B) are consistently defined:

A) P(A) = 0.5, P( B)=0.7, P(A ∩ B)= 0.6.

B) P(B)= 0.5 P(B)= 0.4, P(A U B)= 0.85.

13) Events E and F are such that P(not E or not F)= 0.25, State whether E and F are mutually exclusive.

14) A, B, C are three mutually exclusive events associated with a random experiment. Given P(B)= 3/2 P(A), P(C)= 1/2 P(B), 

find P(A).                                     4/13

15) A, B, C are events such that P(A)= 0.3, P(B)= 0.4, P(C)= 0.8, P(A ∩ B) = 0.08, P(A ∩ C) = 0.28, P(A ∩ B∩ C)= 0.09. if P(A U B UC)≥ 0.75, then show that P(B ∩ C) lies in the interval (0.23, 0.48).

16) The probability of two events A and B are 0.25 and 0.50 respectively. The probability of their simultaneously occurance is 0.14. find the probability that neither A nor B occurs.

17) There are three events A, B and C one of which must and only one can happen, the odds are 8 to 3 against A, 5 to 2 against B, find the odds against C.                       43:34

18) In a race, the odds in favour of horses A, B, C , ad are 1:3,1:4,1:5,1:6 respectively. Find probability that one of them wins the race. 319/420

19) in an easy competition, the odds in favour of competition P, Q, R, S are 1:2,1:3,1:4,1:5 respectively. Find the probability that one of them wins the competition.     114/120

20) A card is drawn at random from a well shuffled deck of 52 cards. Find the probability of its being a spade or a king.                         4/13

21) A card is drawn from a deck of 52 cards. Find the probability of getting an ace or a spade card.    4/13

22) A card is drawn from a deck of 52 cards. Find the probability of getting spade or a king.            4/13

23) Four cards are drawn from a deck of 52 cards. Find the probability that all the drawn cards are of the same colour.         92/883

24) Two cards is drawn from a deck of 52 cards. Find the probability that either both are black or both are kings.                              55/211

25) Two card are drawn from a deck of 52 cards. Find the probability that 2 cards drawn are either aces or black cards.      55/21

26) A card is drawn from a pack of 52 cards. Find the probability of getting a king or a heart or a red card.                                             7/13

27) Four cards are drawn at a time from a pack of 52 cards. Find the probability of getting all the four cards of the same suit.      44/4165

28) Two cards are drawn from a pack of 52 cards. What is the probability that either both are red or both are kings.                 55/221

29) In a single throw of two dice, find the probability that neither a doublet nor a total of 9 will occur.  13/1

30) A die is thrown twice. What is the probability that atleast one of the two throws come up with the number 3 ?                     11/36

31) Find the probability of getting an even number on the first die or a total of 8 in a single throw of two dice.                        5/9

32) A die is thrown twice. What is the probability that atleast one of the two throws comes up with the number 4 ?                             11/36

33) Two dice are thrown together. What is the probability that the sum of the numbers on the two faces is neither divisibile by 3 nor by 4 ?  4/9

34) Two dice are thrown together. What is the probability that the sum of the numbers on the two faces is divisible by 3 or 4?                     5/9

35) A die has two faces with number '1' three faces each with number '2' and one face with number '3'. If the die is rolled once, determine

A) P(1).                                     1/3

B) P(1 or 3).                             1/2

C) P(not 3).                              5/6

36) A natural number is choosen at random from amongst first 500. What is the probability that the number so chosen is divisible by 3 or 5?                       233/500

37) One number is chosen from numbers 1 to 100. Find the probability that it is divisible by 4 or 6.                 33/100

38) An integer is chosen at random from first 200 positive integers. Find the probability that the Integer is divisible by 6 or 8.                 1/4

39) An integer is chosen at random from the numbers ranging from 1 to 50. What is the probability that the Integer chosen is a multiple of 2 or 3 or 10?                               33/50

40) Find the probability of at most two tails or atleast two heads in a toss of three coins.                   7/8

41) One number is chosen from numbers 1 to 200. Find the probability that it is divisible by 4 or 6?                             67/200

42) 100 students appeared for two examinations. 60 passed the first, 50 passed the second and 30 passed both. Find the probability that a student selected at random has passed atleast one examination.                             4/5

43) A box contains 10 white, 6 red and 10 black balls. A ball is drawn at random from the box, what is the probability that the ball drawn is either white or red ?                  8/13

44) A box contains 6 red, 4 white and 5 black balls. A person draws 4 balls from the box at random. Find the probability that among the balls drawn there is atleast one ball of each colour.                           48/91

45) The probability that a person will travel by plane is 3/5 and that he will travel by train is 1/4. What is the probability that he(she) will travel by plane or train.          17/20

46) A box contains 30 bolts and 40 nuts. Half of bolts and half of the nuts are rusted. If two items are drawn at random, what is the probability that either both are rusted or both are bolts.

47) A drawer contains 50 bolts and 150 nuts. Half of the bolts and half of the nuts are rusted. If one item is chosen at random, what is the probability that it is rusted or a bolt ?                                     5/8

48) Find the probability of getting 2 or 3 tails when a coin is tossed four times.                      5/8

49) In an entrance test that is graded on the basis of two examinations, the probability of a randomly selected student passing the first examination is 0.8 and the probability of passing the second examination o.7. the probability of passing atleast one of them is 0.95. What is the probability of passing both.                  0.55

50) The probability that a student will pass the final examination in both English and Hindi is 0.5 and the probability of passing neither is 0.1. if the probability of passing the Hindi examination.                  0.65

51) A basket contains 20 apples and 10 oranges out of which 5 apples and 3 oranges are defective. If a person takes out 2 at random what is the probability that either both are apples or both are good.      316/435

52) The probability that a person will get an electric contact is 2/5 and the probability that he will not get plumbing contract is 4/7. If the probability of getting atleast one contract is 2/3, what is the probability that he will get both ?       17/105

53) The probability that a student will receive A, B, C or D grade are 0.40, 0.35, 0.15 and 0.10 respectively. Find the probability that a student will receive

A) B or C grade.                         0.50

B) at most C grade.                  0.25

54) Let A, B and C be three events. If the probability of occuring exactly one event out of A and B is 1 - x, out of B and C is 1 - 2x, out of C and A is 1 - x, and that of occuring three events simultaneously is x², then prove that the probability that atleast one out of A,B, C will occur is greater than 1/2.   

55) The probability that a patient visiting a dentist will have a tooth extracted is 0.6, the probability that he will have a cavity filled is 0.2 and the probability that he will have a tooth extracted as well as cavity filled is 0.03. what is the probability that a patient has either a tooth extracted or a cavity filled?        0.23

56) The probability that a patient visiting a dentist will have a tooth cleaned is 0.44, the probability that he will have a cavity filled is 0.24 and the probability that he will have a tooth cleaned as well as cavity filled is 0.60. what is the probability that a patient has either a tooth cleaned or a cavity filled?         0.08

57) Probability that Ram passes in mathematics is 2/3 and the probability that he passes in English is 4/9. If the probability of passing both courses is 1/4, what is the probability that Ram will pass in atleast one of these subjects?            31/36

58) In a town of 6000 people 1200 are over 50 years old and 2000 are female. It is known that 30% of the females are over 50 years. What is the probability that a random chosen individual from the town either female or over 50 years.     13/30

59) Two students Ram and Shyam appeared in an examination. The probability that Ram will qualify the examination is 0.05 and that of shyam will qualify the examination is 0.10. the probability that both will qualify the examination is 0.02. find the probability that:

A) both Ram and Shyam will not qualify the exam.                    0.87

B) atleast one of them will not qualify.                                    0.98

C) only one of them will qualify the exam.                            0.11

60) In class XII of a school, 40% of the students study Mathematics and 30% study biology. 10% of the class study both mathematics and biology. If a student is selected at random from the class, find the probability that he will be studying mathematics or biology or both. 3/5

61) In a class of 60 students 30 played football, 32 played cricket and 24 played both football and cricket. If one of these students is selected at random, find the probability that:

A) the student played for football or cricket.                                     19/30

B) the student has played neither football nor cricket.                11/30

C) the student has played football but not cricket.                        2/15

CONDITIONAL PROBABILITY

Let A and B be two events associated with a random experiment. Then, the probability of occurrence of event A under the condition that B has already occurred and P(B)≠ 0, is called the conditional probability and it is denoted by P(A/B). Thus, 

P(A/B)= probability of occurrence of A given that B has already occurred.

Similarly,

P(B/A) when P(A) ≠ 0 is defined as the probability of occurrence of event B when A has already occurred.

The meanings of symbols P(A/B) and P(B/A) depend on the nature of the events A and B and also on the nature of the random experiment. 

These two symbols have the following meaning also.

P(A/B)= Probability of occurrence of A when B occurs.

OR

Probability of occurrence of A when B is taken as the sample space.

OR

Probability of occurrence of A with respect to B

AND

P(B/A)= Probability of occurrence of B when A occurs.

OR

Probability of occurrence of B when A is taken as the sample space.

OR

Probability of occurrence of B with respect to A

Example:1

 Let there be a bag containing 5 white and 4 red balls. Two balls are drawn from the bag one after the other without replacement. 

Consider the following events:

A= Drawing a white ball in the first draw.

B= Drawing a red ball in the second draw.

P(B/A)= Probability of drawing a red ball in second draw given that a white ball has already been drawn in the first draw.

P(B/A)= Probability of drawing a red ball from a bag containing 4 white and 4 red balls.

P(B/A)= 4/8= 1/2

Here P(A/B) is not meaningful because A can not occur after the occurrence of event B.

Example: 2

Consider the random experiment of throwing a pair of dice and two events associated with it given by

A= The sum of the numbers on two dice is 8= {(2,6),(6,2),(3,5),(5,3),(4,4)}

B= There is an even number on first die={(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(6,1),(6,2),(6,3),(6,4),(6,5)(6,6)}.

P(A/B)= Probability of occurrence of A when B occurs

OR

Probability of occurrence A when B is taken as the sample space.

P(A/B)=(Number of elementary events in B which are favorable to A)/(Number of elementary events in B)

OR

P(A/B)=(Number of elementary events favorable to (A∩B)/(Number of elementary favourable to B) = 3/18

Similarly 

P(B/A)= Probability of occurrence of B when A occurs.

OR

Probability of occurrence B when A is taken as the sample space.

P(B/A)=(Number of elementary events in A which are favorable to B)/(Number of elementary events in A)

OR

P(B/A)=(Number of elementary events favorable to (A∩B)/(Number of elementary favourable to A) = 3/5

Example 3:

A die is thrown twice and the sum of the numbers appearing is observed to be 6. What is the conditional probability that the number 4 has appeared atleast once?

Consider the following events:

A= Number 4 appears atleast once.

B= sum of the numbers appearing is 6.

Required probability= 

P(A/B) = Probability of occurrence of A when B is taken as the sample space.

OR

P(A/B)=(Number of elementary events favorable to A which are favourable to B)/(Number of elementary events in B)

OR

P(A/B)=(Number of elementary events favorable to (A∩B)/(Number of elementary favourable to B) = 2/5

 OR

In short

P(A/B)= (Number of elementary events favorable to (A∩B)/(Number of elementary favourable to B) 

OR

P(A/B)= n(A∩B)/n(B)

         = n(A∩B)/n(S)/n(B)/n(S)

OR

P(A/B)= n(A∩B)/n(B)

Similarly

P(B/A)= P(A∩B)/P(A)

Formula:

1) P(A)= P(A∩B)+ P(A∩B')

2) P(B)= P(A∩B)+ P(A'∩B)

3) P(A U B) = P(A∩B)+ P(A∩B') + P(A'∩B).

4) P(A/B)= n(A∩B)/n(B)

5) P(B/A)= n(A∩B)/n(A)

EXERCISE - D

1) A four dice is rolled consider the following events:

A={1,3,5}, B={2,3} and C={2,3,4,5} find

A) P(A/B).                                   1/2

B) P(B/A).                                   1/3

C) P(A/C).                                   1/2

D) P(C/A).                                   2/3

E) P(A UB/C).                             3/4

F) P(A∩B/C).                              1/4

2) A coin is tossed three times. Find P(E/F) in each of the following:

i) E= Head on third toss, F= Heads on first two tosses.                  1/2

ii) E= Atleast two heads, F= atmost two heads.                                3/7

iii) At most two tails, F= atleast one tail.                                              6/7

3) Two coins are tossed once. Find P(E/F) in each of the following:

i) E= Tail appears on one coin, F= One coin shows head.                 1

ii) No tail appears, F= No head appears.                                         0

4) Mother, father and son line up at random for a family picture. Find P(AB), if A and B are defined as follows:

A= Son on one end.

B= Father in the middle.                1

5) A and B are two events such that P(A)≠ 0. Find P(B/A), if

A) A is a subset of B.                      1

B) A∩B = nul set.                            0

6) Given that A and B are two events such that P(A)= 0.6, P(B)= 0.3 and P(A∩B) = 0.2, find

A) P(A/B).                                2/3

B) P(B/A).                              1/3

7) If P(A)= 6/11, P(B)= 5/11 and P(A U B)= 7/11, find

A) P(A∩B).                                4/11

B) P(A/B).                                  4/5

C) P(B/A).                                 2/3

8) Evaluate P(A U B), If 2P(A)= P(B) = 5/13 and P(A/B)= 2/5.       11/26

9) If P(B)= 0.5 and P(A∩B) = 0.32 find P(A/B).                         16/25

10) If P(A)= 0.4, P(B)= 0.3 and P(B/A)= 0.5, find

A) P(A∩B).                                0.2

B) P(A/B).                                 2/3

11) If A and B are two events such that P(A)= 1/3, P(B)= 1/5 and P(A UB)= 11/30. Find

A) P(A/B).                               5/6

B) P(B/A).                               1/2

12) A basket contains 20 apples and 10 oranges out of which 5 apples and 3 oranges are defective. If a person takes out 2 at random what is the probability that either both are apples or both are good ?                316/435 

13) The probability that a person will get an electric contract is 2/5 and the probability that he will not get plumbing contract is 4/7. If the probability of getting atleast one contract is 2/3, what is the probability that he will get both ?           17/105

14) Ten cards numbered 1 through 10 are placed in a box, mixed up thoroughly and then one card is drawn and drawn randomly. if it is known that the number on the drawn card is more than 3, what is the probability that it is an even number ?                                4/7

15) A pair of dice is thrown. If the two numbers appearing on them are different. Find the probability

A) the sum of the numbers is 6.    2/15

B) the sum of the numbers is 4 or less.                     1/15

C) the sum of the numbers is 4.      2/15

16) A dice thrown three times, find the probability that 4 appears on the third toss if it is given that 6 and 5 appear respectively on first two tosses.                      1/6

17) A die is thrown three times. Events A and B are defined as below:

A= Getting 4 on third die,

B= Getting 6 on the first and 5 on the second throw.

Find the probability of A given that B has already occurred.              1/6

18) A die is thrown twice and the sum of the numbers appearing is observed to be 6. What is the probability that the number 4 has appeared atleast once?            2/5

19) A die is thrown three times. If the first throw is a four, find the probability of getting 15 as the sum.                                     1/18

20) A coin tossed three times, if head occurs on first two tossed, find the probability of getting head on third toss.                            1/2

21) A black and a red dice are rolled in order. Find the conditional probability of getting

A) a sum greater than 9, given that the black die resulted in a 5. 1/3

B) a sum 8, given that the red die resulted in a number less than 4.      1/9

22) Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the die again and if any other number comes, toss a coin. Find the conditional probability of the event 'the coin shows a tail ' given that atleast one die shows a three.                        0

23) Assume that each child born is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that 

A) the youngest is a girl.           1/3

B) atleast one is a girl.              1/3

24) Given that the two numbers appearing on throwing two dice are different. Find the probability of the event 'the sum of numbers on the dice is 4'.                                 1/15

25) A couple has two children. Find the probability that both the children are

A) males, if it is known that at least one of the children is male.

B) females, if it is known that the elder child is the female.     1/3, 1/2

26) A couple has two children. Find the probability that 

A) both the children are boys, if it is known that the older child is a boy.                 1/2

B) both the children are girls, if it is known that the older child is a girl.                        1/2

C) both the children are boys, if it is known that atleast one of the children is a boy.                       1/3

27) In a school, there are 1000 students, out of which 430 are girls. It is known that out of 430, 10% of the girls study in class XII. What is the probability that a student selected randomly studies in class XII given that the chosen student is a girl.                     1/10

28) Two integers are selected at random from integers 1 to 11. If the sum is even, find the probability that both the numbers are odd.  3/5

FORMULA:

If A and B are two events associated with a random experiment, then 

1) P(A∩B)= P(A) P(B/A), if P(A)≠0

OR

P(A∩B)= P(B) P(A/B), if P(B)≠0

2) If A, B, C are three events associated with a random experiment, then

P(A∩B∩C)= P(A) P(B/A) P(C/A∩ B).

3) P(A UB)= 1 - P(A') P(B')

Exercise - E

1) From a pack of 52 cards, two are drawn one by one without replacement. Find the probability that the both of them are kings.    1/221

2) From a pack of 52 cards, 4 are drawn one by one without replacement. Find the probability that all are aces(or Kings).     1/270725

3) From a deck of cards, three cards are drawn one by one without replacement. Find the probability that each time it is a card of spade.         11/850

4) Two cards are drawn without replacement from a pack of 52 cards. Find the probability that

A) both are kings.                    1/221

B) the first is a king and the second is an ace.                                 4/663

C) the first is a heart and second is red. 25/204

5) A card is drawn from a well shuffled deck of 52 cards and then a second card is drawn. Find the probability that the first card is a heart and the second card is a diamond if the first card is not replaced.                          13/204

6) Three cards drawn successfully, without replacement from a pack of 52 well shuffled cards. What is the probability that first two cards are kings and third card drawn is an ace ?                    2/5525

7) Find the probability of drawing a diamond card in each of the two consecutive draws from a well shuffled pack of cards, if the card drawn is not replaced after the first draw.            1/17

8) Find the chance of drawing two white balls in succession from a bag containing 5 red and 7 white balls, the balls first drawn not being replaced.                  7/22

9) A bag contains 5 white, 7 red and 8 black balls. If balls are drawn one by one without replacement, Find the probability of getting all white balls.               1/969

10) A bag contains 5 white and 8 black balls. Two successive drawings of three balls are a time are made such that the balls are not replaced before the second draw. Find the probability that the first draw gives 3 white balls and second draw gives 3 black balls.           7/429

11) An urn contains 3 white, 4 red and 5 black balls. Two balls are drawn one by one without replacement. What is the probability that atleast one ball is black.  15/22

12) A bag contains 5 white, 7 red and 3 black balls. If three balls are drawn one by one without replacement, Find the probability that none is red.                       8/65

13) An urn contains 10 black and 5 white balls. Two balls at drawn from the urn one after the other without replacement. What is the probability that both drawn balls are black ?                                  3/7

14) A bag contains 4 white, 7 black and 5 red balls. 3 balls are drawn one after the other without replacement. Find the probability that the balls drawn are white, black and red respectively.               1/24

15) A bag contains 10 white and 15 black balls. Two balls are drawn in succession without replacement. What is the probability that first is white and second is black.          1/4

16) A bag contains 19 numbered from 1 to 19. A ticket is drawn and then another ticket is drawn without replacement. Find the probability that both tickets will show even numbers.                                4/19

17) Two balls are drawn from an urn containing 2 white, 3 red and 4 black balls one by one without replacement. What is the probability that atleast one ball is red.       7/12

18) A bag contains 25 tickets, numbered from 1 to 25. A ticket is drawn and then another ticket is drawn without replacement. Find the probability that both tickets will show even numbers.              11/50

19) A bag contains 20 tickets numbered from 1 to 20. Two tickets are drawn without replacement. What is the probability that the first ticket has an even number and second an odd number.           5/19

20) A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three orange are good, the box is approved for sale otherwise it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.                    44/91

21) To test the quality of electric bulbs produced in a factory, two bulbs are randomly selected from a larger sample without replacement. If either bulb is defective, the entire lot is rejected. Suppose a sample of 200 bulbs contains 5 defective bulbs. Find the probability that the sample will be rejected.  197/3980

Exercise - F

1) If P(A)= 7/13, P(B)= 9/13 and P(A∩B)= 4/13, find P(A/B).     4/9

2) If A and B are events such that P(A)=0.6, P(B)=0.3 and P(A∩B)=0.2, find

A) P(A/B).                         2/3

B) P(B/A).                        1/3

3) if A and B are two events such that P(A∩B)= 0.32 and PB =0.5, find P(A/B).                   0.64

4) If P(A)=0.4 , P(B)=0.8, P(B/A)=0.6

Find

A) P(A/B).                     0.3

B) P(A U B).                 0.96

5) If A and B are two events such that

A) P()=1/3, P(B)=1/4 and P(AUB)=5/12 find

a) P(A/B).                            2/3

b) P(B/A).                           1/2

B) P(A)==6/11, P(B)= 5/11 and P(AUB)=7/11, find

a) P(A∩B).                      4/11

b) P(A/B).                        4/5

c) P(B/A).                        2/3

C) P(A)=7/13, P(B)=9/13 and P(A∩B)=4/13, find P(A'/B).      5/9

D) P(A)=1/2, P(B)=1/3 and P(A∩B)= 1/4, find

a) P(A/B).                      3/4

b) P(B/A).                     1/2

c) P(A'/B).                     1/4

d) P(A'/B').                     5/8

6) If A and B are two events such that 2P(A)=P(B)= 5/13 and P(A/B)= 2/5, find P(AUB).                  11/26

7) If P(A)=6/11, P(B)=5/11 and P(AUB)=7/11, find 

A) P(A∩B).                               4/11

B) P(A/B).                                  4/5

C) P(B/A).                                   2/3

8) If A and B are two events such that P(A)=0.5, P(B)=0.6 and P(A U B) = 0.8, find

A) P(A/B).                               1/2

B) P(B/A).                               3/5

9) If A and B are two events such that P(A)=0.3, P(B/A)=0.5 Find 

A) P(AUB).                                 0.75

B) P(A/B).                                    1/4

10) If P(not A)=0.7, P(B)=0.7 and P(B/A)= 0.5, then find

A) P(AUB).                                 0.85

a) P(A/B).                                   3/14

11) If A and B are two events associated with a random experiment such that P(A)=0.8, P(B)=0.5 and P(B/A)=0.4 find

A) P(A∩B).                                  0.32

a) P(A/B).                                    0.64

b) P(A U B).                                 0.98

12) A fair die is thrown. Consider the events

A= {1,3,5}, B={2,3}, C={2,3,4,5}. Find

A) P(A/B).                                 1/2

B) P(B/A).                                  1/3

C) P(A/C).                                   1/2

D) P(C/A).                                   2/3

E) P(B/C).                                    1/2

E) P(AUB/C).                                3/4

F) P(A∩B/C).                                1/4 

13) Three events A, B and C have Probabilities 2/5, 1/3 and 1/2 respectively. Given that P(A∩C)= 1/5 and P(B∩C)= 1/4, find the values of 

A) P(C/B).                                  3/4

B) P(A'∩C').                               3/10

14) If P(A)= 3/8, P(B)= 1/2 and P(A∩B)=1/4 find

A) P(A'/B').                                 3/4

B) P(B'/A').                                 3/5

15) A coin is tossed three times. Find P(A/B) in each case of the following:

a) A= Head on 3rd toss, B= heads on first two tosses.                   1/2

b) A= at least two heads, B= atmost two heads.                                  3/7

c) A= at most two tails, B= at least one tail.                                        6/7

16) A coin is tossed twice and the four possible outcomes are assumed to be equally likely. If A is the event, 'both head and tail have appeared ', and B be the event, 'at most one tail is observed ', find 

A) P(A).                                         1/2

B) P(B).                                         3/4

C) P(A/B).                                      2/3

D) P(B/A).                                      1

17) Two coins tossed once. Find P(A/B) in each case of the following:

A) tail appears on one coin, B= One coin shows head.                             1

B) A= no tail appears, B= No head appears.                                            0

18) Two coins are tossed. What is the probability of coming up two heads if it is known that atleast one head comes up.                           1/3

19) A die is thrown three times. Find P(A/B) and P(B/A), if 

A) 4 appears on the third toss, B= 6 and 5 appear respectively on 1st two tosses.                         1/6, 1/36

20) A dice is thrown twice and the sum of the numbers appearing is observed to be six. What is the condition of probability that the number 4 has appeared atleast once.                            2/5

21) A dice is thrown. If outcome is an odd number, what is the probability that it is prime.         2/3

22) A dice thrown twice and the sum of the numbers appearing is observed to be 8, what is the conditional probability that the number 5 has appeared at least once.                       2/5

23) A dice thrown twice and the sum of the numbers appearing is observed to be 7, what is the conditional probability that the number 2 has appeared at least once.                     1/3

24) A die is thrown three times. Events A and B are defined as follows:

A: 4 on third throw, B: 6 on the first and 5 on the second throw, find the probability of A given that B has already occurred.                 1/6

25) two dice are thrown. Find the probability that the number appeared has the sum 8, if it is known that the second die always exhibits 4.                          1/6

26) A pair of dice is thrown. Find the probability of getting 7 and the sum, If it is known that second dice always exhibits an odd number. 1/6

27) A pair of dice is thrown. Find the probability of getting 7 as the sum if it is known that the second die always exhibits a Prime number.              1/6

28) A pair of dice is thrown. Find the probability of getting the sum 8 or more, if 4 appears on the first die.                      1/2

29) Two dice are thrown and it is known that the first die shows a 6. Find the probability that the sum of the numbers showing on two dice is 7.                      1/6

30) A pair of dice is thrown. Let E be the event that the sum is greater than or equal to 10 and F be the event 5 appears on the first die. Find P(E/F). If is the event 5 appears on at least one die. find P(E/F).                       1/3, 3/11

31) Three dice are thrown at the same time. Find the probability of getting three two's if it is known that the sum of the numbers on the dice was a six.                   1/10

32) A black and a red die rolled.

A) find the conditional probability of getting a sum greater than 9, given that the black die resulted in a 5.     1/3

B) Find the conditional probability of getting the sum of 8, given that the red die resulted in a number less than 4.                     1/9

33) Find the probability the sum of the numbers is showing on two dies is 8, given that at least one die does not show 5.                     3/25

34) two numbers are selected at random from Integers 1 through 9. If the sum is even, find the probability that both the numbers are odd.                   5/8

35) Two Integers are selected at random from integers 1 through 11. If the sum is even, find the probability that both the numbers are odd.                     3/5

36) 10 cards numbered 1 to 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is a more than three. What is the probability that it is an even number ?              4/7

37) the probability that a student selected at random from a class will pass in the Mathematics is 4/5, and the probably that/he/she is passes in mathematics and computer science is 1/2. What is the probability that he/she will pass in computer science if it is known that he/she has passed in mathematics?                     5/8

38) The probability that a certain person will buy a shirt is 0.2, the probability that he will buy a trouser is 0.3, and the probability that he will buy a shirt given that he buys a trouser is 0.4. Find the probability that he will buy both a shirt and a trouser. 

Find also the probability that he will buy a trouser given that buys a shirt.                         0.12, 0.6

39) In a school there are 1000 students, out of which 430 are girls. It is known that out of 430, 10% the girls study in class XII. What is the probably that a student selected randomly studies in class XII given that chosen student is a girl ?   1/10

40) Assume that is each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls? Given that

A) the youngest is a girl.             1/2

B) at least one is girl.                  1/3

41) A couple has 2 children. Find the probability that Both are boys, if it is known that

A) one of the children is a boy.   1/3

B) the older child is a boy.           1/2

42) Mother father and son line up at random for a family picture. If A and B are two events given by A= son on one end, B= father in the middle, find 

A) P(A/B).                            1

B) P(B/A).                           1/2

43) 10% of the bulbs produced in a factory are red colour and 2% are red and defective. If one bulb is picked at random, determine the probability of its being defective if it is red.                          1/5

44) Consider a random experiment in which a coin is tossed and if the coin shows head it is tossed again but if it shows a tail then a die is tossed. If 8 possible outcomes are equally likely, find the probability that the die shows a number greater than 4 if it is known that the first throw of the coin results in a tail.                    1/3

45) A bag contains 3 red and 4 black balls and another bag has 4 red and 2 black balls. One bag is selected at random and from the selected bag a ball is drawn. Let A be the event that the first bag is selected, B be the event that the second bag is selected and C be the event that the ball drawn is red. Find

A) P(A).                    1/2

B) P(B).                    1/2

C) P(C/A).                3/7

D) P(CB).                  2/3

46) Three distinguishable balls are distributed in three cells. Find the conditional probability that all the three occupy the same cell, given that atleast two of them are in the same cell.                        1/7

47) A coin is tossed, then a die is thrown. Find the probability of getting a 6 given that head came up.                              1/6

48) Consider the experiment of tossing a coin. If the coin shows head loss it again but if it shows tail then throw a dice. Find the conditional probability of the event the die shows a number greater than 4, given that there is atleast one tail.                    2/9 or 2/7

49) Consider the experiment of throwing a die, if a multiple of 3 comes up throw the die again and if any other number comes toss a coin. Find the conditional probability of the event 'the coin shows a tail ', given that atleast one die shows a 2.              3/8 or 1/4

50) A committee of 4 students is selected at random from a group consisting of 8 boys and 4 girls. Given that there is atleast one girl in the committee, calculate the probability that there are exactly 2 girls in the committee.       168/425

51) In a hostel 60% of the students read Hindi newspaper, 40% read English newspaper and 20% read both Hindi and English newspapers. A student is selected at random. Find

A) the probability that she reads neither Hindi nor English newspapers.                          1/5

B) If she reads Hindi newspaper, find the probability that she reads English newspapers.                1/3

C) if she reads English newspaper, find the probability that she reads Hindi newspaper.                        1/2

52) An electronic assembly consists of two sub-systems say A and B. From previous testing procedures, the following probabilities are assumed to be known.

P(A fails)= 0.2, P(B fails alone)= 0.15, P(A and B fail)= 0.15

Evaluate the following probabilities:

A) P(A fails/B has failed).          1/2

B) P(A fails alone).                   0.05

____________

Formula:

1) If A and B are independent events associated with a random experiment, then

P(A∩B)=P(A) P(B).

2) If A and B are independent events associated with a random experiment, then

i) A' and B are independent.

ii) A and B ' are independent events.

iii) A' and B ' are also independent events.

Exercise - G

1) A coin is tossed thrice and all the eight outcomes are assumed equally likely. In which of the following cases are the following events A and B are independent?

i) A= the First throw results in head,

  B= the last throw results in tail.    Y

ii) A= the number of heads is odd,

 B= the number of tails is odd.

iii) A= the number of heads is two

 B= the last throw results in head.

2) A lot contains 50 defective and 50 non-defective bulbs. Two bulbs are drawn random, one at a time, with replacement. The events A, B, and C are defined as

A: the first bulb is defective

B: the second bulb is non defective.

C: the two bulbs are both defective or both non defective.

Determine whether

i) A, B and C are pairwise independent.

ii) A, B and C are mutually independent.       

3) A card is drawn from a pack of 52 cards so that each card is equally likely to be selected. In which of the following cases are the events A and B independent?

i) A= the card drawn is a king or queen,

B= the card drawn is a queen or Jack.

ii) A= the card drawn is black,

 B= the card drawn is a king.          Y

iii) B= the card drawn is a spade,

 B= the card drawn in an ace.         X

4) A coin is tossed thrice and all eight outcomes are equally likely.

E: The first throw results in head.

F: The last throw results in tail.

Prove that events E and F are independent.

5) A coin is tossed three times. Let the events A, B and C be defined as follows:

A= first toss is head, B= second toss his head, C= exactly two heads are tossed in a row.

 Check the independence of

i) A and B .                                         I

ii) B and C.                                         D

iii) C and A.                                        I

6) an unbiased dice is thrown twice. Let the event A be odd number on the first throw and B the event odd number on the second throw. Check the independent of events A and B.

7) If A and B be two events such that P(A)= 1/4, P(B)= 1/3 and P(A U B)= 1/2, show that A and B are independent events.

8) Two dice are thrown together and the total score is noted. The events E, F and G are a total 4, a total of 9 or more, and a total divisible by 5, respectively. Calculate P(E), P(F) and P(G) and decide which pair of events, if any, are independent.

9) Three coins are tossed. consider the event:

E= 3 heads or 3 tails,

F= at least two heads 

G= Atmost two heads.

of the pairs (E, F), (E,G) and (F,G) which are independent ? Which are dependent ?    

10) A fair coin and an unbiased die are tossed. Let A be the event 'head appayers on the coin' and B be the event '3 on the die' check whether A and B are Independent or not.       Y

11) A die is marked 1, 2, 3 in red and 4, 5 ,6 in green is tossed. Let A be the event 'nunber is even' and B be the event 'numer is red. Are A and B independent.                       N

12) Event A and B are such that P(A)= 1/2, P(B)=7/12 and P( not A or not B)= 1/4. State whether A and B are Independent?                          N

13) A die if thrown once. If A is the event 'the number appearing is a multiple of 3' and B is the event ' the number appearing is even'. Are the events A and B independent ?         Y

14) Two dies are thrown together. let A be the event 'getting 6 on the first die' and B be the event 'getting 2 on the second die'. Are the events A and B independent ?                   Y

15) For a loaded die, the probabilities of outcomes are given as under:

P(1)=P(2)= 2/10, P(3)= P(5)= P(6) = 1/10 and P(4)= 3/10.

The die is thrown two times. Let A and B be the events as defined below

A= getting same number each time,

B= getting a total score of 10 or more.

Determine whether or not A and B are independent events 

16) In the above question, if the die were fair, determine whether or not the events A and B are independent.         N

17) In the two dice experiment, if A is the event of getting the sum of the numbers on dice as 11 and B is the event of getting a number other than 5 on the first die, find

P(A and B) are A and B independent events?                  N

18) Given two independent events A and B such that P(A)= 0.3 and P(B)= 0.6, find

A) P(A∩B).                                0.18

B) P(A∩B').                               0.12

C) P(A'∩B).                               0.42

D) P(A'∩B').                              0.28

E) P(AUB).                                0.72

F) P(A/B).                                  0.3

G) P(B/A).                                 0.6

19) If P(not B)= 0.65,P(AUB)= 0.85 and A and B are independ events, then find P(A).                            0.77

20) if A and B are two Independent events such that P(A'∩B)=2/15 and P(A∩B')= 1/6, then find P(B).       1/6 or 4/5

21) A and B are two Independent events. The probability that A and B occur is 1/6 and the probability that neither of them occurs is 1/3. Find the probability of a occurrence of two events.       1/3 , 1/2 or 1/2, 1/3

22) If A and B are two independent events such that P(AUB)=0.60 and P(A)=0.2, find P(B).                      0.5

23) Given that the events A and B are such that P(A)=1/2, P(AUB)= 3/5 and P(B)= p. find p, if they are

A) mutually exclusive.              1/10

B) independent .                         1/5

24) if A and B are two events such that P(A)= 1/4, P(B)= 1/2 and P(A∩B)=1/8, find P(not A and not B).                         3/8

25) events E and F are independent find 

A) P(F), if P(E)= 0.35 and P(E F)= 0.6 .                    5/13

26) P(A)= 0.4, P(B)= p, P(AUB)=0.6 and And B are given to be independent events, find the value of P.                    1/3

27) let A and B be two independent events. The probability of their simultaneous occurrence is 1/8 and the probability that neither occurs is 3/8. Find 

A) P(A) .        

B) P(B).            1/2, 1/4 or 1/4, 1/2

28) if A and B are two independent events such that P(A'∩B)=2/15 and P(A∩B')=1/6, then find

A) P(A)

B) P(B).              1/5, 1/6 or 5/6, 4/5

29) A dice is tossed twice. Find the probability of getting a number greater than 3 on each toss.       1/4

30) An unbiased die is tossed twice. Find the probability getting 4, 5, or 6 on the first toss and 1, 2, 3 or 4 on the second toss.                 1/3

31) A die is thrown thrice. Find the probability of getting an odd number at least once.               7/8

32) two dice are thrown. Find the probability of getting an odd number on the first die and a the multiple of 3 on the other.         1/6

33) In two successive throws of a pair of dice, determine the probability of getting a total of 8 each time.                            25/1296

34) A bag contains 3 red and 2 black balls. One ball is drawn from it at random. Its colour is noted and then it is put back in the bag. A second draw is made and the same procedure is repeated. Find the probability of drawing 

A) two red balls.                          9/25

B) two black balls.                     4/25

C) first red and second block ball.      6/25

35) Two balls are drawn at random with replacement from a box containing 10 blacks and 8 red balls. Find the probability that

A) both balls are red.        16/81

B) first ball is black and second is red.                                 20/81

C) one of them is black and other is red.                   40/81

36) An urn contains 4 red and 7 black balls. Two balls are drawn at random with replacement. Find the probability of getting

A) two red balls.                    16/121

B) two black balls.                 49/121

C) 1 red and one black.         56/121

37) A bag contains 5 white.7 red and 4 Black balls. If 4 balls a drawn one by one with replacement, what is the probability that none is white ?               (11/16)⁴

38) ) a bag contains 3 red and 5 black balls and a second bag contains 6 red and 4 Black balls. A ball is drawn from each bag. Find the probability that both are 

A) red.                    6/10

B) black.                 9/40

39) A bag contains 5 white,7 red and 8 black balls. 4 balls are drawn one by one with replacement, what is the probability that at least one is white ?                 1 -(3/4)⁴

40) Three cards are drawn with replacement from a well shuffled pack of cards. Find the probably that the cards drawn are king, queen and jack.                 6/2197

41) ) A can hit a target 4 times in 5 shots, B 3 times in 4 shots, and C 2 times in 3 shots. calculate the probability that 

A) A, B, C all may hit.                    2/5

B) B, C may hit and A may not.  1/10

C) any two of A ,B and C will hit the target.                     13/30

D) none of them will hit the target.         1/60

42) The probability that A hits a target is 1/3 and the probably that B hits it, is 2/5. What is the probability that the target will be hit, if each one of A and B shoots at the target ?          3/5

43) Given the probability that A can solve a problem is 2/3 and the probability that B can solve the same problem 3/5. Find the probability that none of the two will be able to solve the problem.   2/15

44) ) A problem in mathematics is given to 3 students whose chances of solving it are :1/2, 1/3, 1/4. What is the probability that problem is solved.                  3/4

45) A can solve 90% of the problems given in a book and B can solve 70%. What is the probability that at least one of them will solve the problem, selected at random from the book ?               0.97

46) ) probabilities of solving a specific problem independently by A and B are 1/2 and 1/3 respectively. If both try to solve the problem independently, find the probability that.

A) the problem is solved.            2/3

B) exactly one of them solve the problem.                   1/2

47) A and B are candidate seeking admission in a college. The probability that A is selected is 0.7 and the probability that exactly one of them is selected is 0.6. Find the probability that B is selected.     1/4

48) An article manufactured by a company consist of two parts X and Y. In the process of manufacture of the part X, 9 out of 100 parts may be defective. Similarly, 5 out of 100 are likely to be defective in the manufacture of part Y. Calculate the probability that the assembled product will not defective.                       0.8645

49) An anti-craft gun can take a maximum of 4 shots at an enemy plane moving away from it. The probability of hitting the plane at the first, second, third and fourth shot are 0.4, 0.3, 0.2 and 0.1 respectively. What is the probability that gun hits the plane.           0.696

50) ) The probability of two students A and B coming to the school in time are 3/7 and 5/7 respectively. Assuming that the events, 'A coming in time and 'B coming in time are independent. Find the probability of only one of them coming to the school in time. Write at least one advantage of coming to school in time.      26/49

51) A class consists of 80 students, 25 of them are girls and 55 boys, 10 of them are rich and the remaining poor, 20 of them are fair complexioned. What is the probability of selecting a fair complexioned rich girl ?       5/512

52) policeman fires four bullets on a dacoit. The probability that the dacoit will be killed by any Bullet is 0.6. What is the probability that the dacoit is still alive ?             0.0256

53) The probably that a teacher will give an urn announced test during any class meeting is 1/5. if a student is absent twice, what is the probability that he will miss at least one test ?                    9/25

54) A machine operates if all of its three components function. The probability that the first component fails during the year is 0.14, the second component fails is 0.10 and the third component fails is 0.05. What is the probability that the machine will fail during the year?       0.2647

55) A scientist has to make a decision on each of the two independent events I and II. Suppose the probability of error in a making decision on event I is 0.2 and that an event II is 0.05. Find the probability thst the scientist will make the correct decision on

A) both the events.               0.931

B) only one event.                0.068

56) A town has two fire extinguishing engines functioning independently. The probability availability of each engine, when needed is 0.95. What is the probability that

A) neither of them is available when needed?                      0.0025

B) an engine is available when needed.                      0.9975

C) exactly one engine is available when needed ?            0.095

57) A company has estimated that the probabilities of success for 3 products introduced in the market are 1/3, 2/5 and 2/3 respectively. Assuming independence, find the probability that

A) the three products are successful.                       4/45

B) none of the products is successful.                      2/15

58) The odds against A solving a certain problem are 4 to 3 and the odds in favour of B solving the same problem are 7 to 5. Find that the problem will be solved.     16/21

59) ) The odds against a certain event are 5 to 2 and the odds in favour of another event, independent to the former are 6 to 5. Find the probability that

A) atleast one of the events will occur.                     52/77

B) none of the events will occur.    25/77 

60) A coin is tossed and a dice is thrown. Find the probability that the outcome will be a head or a number greater than 4, or both.              2/3

Exercise - H 

1) A bag contains 4 white and 2 black balls. Another contains 3 white and 5 black balls. If 1 ball is drawn from each bag, find the probability that 

A) both are white. 1/4

B) both are black. 5/24

C) one is white and one is black. 13/24

2) A box contains 3 red and 5 blue balls. 2 balls are drawn one by one at a time at random without replacement. Find the probability of getting 1 red and 1 blue ball.  15/28

3) A bag contains 5 white and 3 black balls. 4 balls are successfully drawn out without replacement. What is the probability that they are alternatively of different colours ? 1/7

4) Bag A contains 4 red and 5 black balls and bag B contains 3 red and 7 black balls. One ball is drawn from bag A and two from bag B. Find the probability that out of 3 balls drawn, two are black and one is red.                      7/15

5) A bag contains 5 red Marbles and 3 black Marbles. Three marbles are drawn one by one without replacement. What is the probability that atleast one of the three marbles drawn be black, if the first marble is red ?                       25/56

6) A bag contains 3 white,3 black and 2 red balls. One by one, three balls are drawn without replaceing them. Find the probability the third ball is red.                           1/4

7) There are three urns A,B and C. Urn A contains 4 white balls and 5 blue balls. Urn B contains 4 white balls and 3 blue balls. Urn C contains 2 white balls and 4 blue balls. One ball is drawn from each of these urns. What is the probability that out of these three balls drawn, two are white balls and one is a blue ball.                 64/189

8) A bag contains 6 black and 3 white balls. Another bag contains 5 black and 4 white balls. If one ball is drawn from each bag, find the probability that these two balls are of the same colour. 14/27

9) A bag contain 3 red and 5 black balls and a second bag contains 6 red and 4 black balls. A ball is drawn from each bag. Find the probability that one is red and the other is black. 21/40

10) Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls Find the probability that

A) both the balls are red. 16/81

B) first ball is black and the second is red. 20/81

C) one of them is black and other is red. 40/81 

11) A bag contains 3 white, 4 red and 5 black balls. Two balls are drawn one after the other, without replacement. What is the probability that one is white and the other is black ?                               5/22

12) A bag contains 8 red and 6 green balls. 3 balls are drawn one after another without replacement. Find the probability that atleast two positive drawn are green.        5/13

13) A bag conta8 7 white, 5 black and 4 red balls. Four balls are drawn without replacement. Find the probability that at least 3 balls are black.                             23/364

14) A bag contains 4 white balls and 2 black balls. Another contains 3 white balls and 5 black balls. If one ball is drawn from each bag, find the probability that

A) both are white. 1/4, 5/24, 13/24

B) both are black.

C) one is white and one is black.

15) A bag contains 4 white, 7 black and 5 red balls. 4 balls are drawn with replacement. What is the probability that atleast two are white ? 67/256

16) A bag contains 4 red and 5 black balls, a second bag contains 3 red and 7 black balls. One ball is drawn at random from each bag, find the probability that the

A) balls are of different colours. 43/90

B) balls are of the same colour. 47/90

17) There are three urns A, B, C. Urn A contains 4 red balls and 3 black balls. Urn B contains 5 red balls and 4 black balls. Urn C contains 4 red and 4 black balls. 1 ball is drawn from each of these urns. What is the probability that 3 balls drawn consist of 2 red balls and a black balls? 17/42

18) A bag contains 8 marbles of which 3 are blue and 5 are red. One marble is drawn at random, its colour is noted and the marble is replaced in the bag. A marble is again drawn from the bag and its colour is noted. Find the probability that the marble will be

A) blue followed by red. 15/64

B) blue and red in any order. 15/32

C) of the same colour. 17/32

19) An urn contains 7 red and 4 Blue Balls. two balls are drawn at random with replacement. Find the probability of getting :

A) two red balls. 49/121

B) 2 Blue Balls. 16/121,56/121

C) one red and one blue ball.

20) There are 3 red and 5 black balls in bag 'A'; and 2 red and 3 black balls in bag 'B'. One ball is drawn from bag 'A' and two from bag 'B'. Find the probability that out of the 3 balls drawn one is red and 2 are black. 39/80

21) A card is drawn from a well shuffled deck of 52 cards. The outcome is noted, the card is replaced and the deck reshuffled. Another card is then drawn from the deck.

A) what is the probability that both the cards are of the same suit ? 1/4, 1/338

B) What is the probability that the first card is an ace and the second card is red Queen.   

22) Two cards are drawn from a pack of 52 cards without replacement. What is the probability that one is red queen and the other is a king of black colour? 2/663

23) two cards are drawn without replacement from a well shuffled pack of 52 cards. Find the probability that one is a spade and other is a queen of red colour. 1/51

24) cards are numbered 1 to 25. Two cards are drawn one after the other. Find the probability that the number of one card is multiple of 7 and on the other it is a multiple of 11. 1/50

25) Two cards and drawn successfully without replacement from a well shuffled deck of 52 cards. Find the probability of exactly one ace. 32/221

26) Two cards a drawn from a well shuffled pack of 52 cards, one after another without replacement. Find the probability that one of these is red card and the other a black card ? 26/51

27) Three cards are drawn with replacement from a well shuffled pack of 52 cards. Find the probability that the cards are a king, a queen and a jack. 6/2197

28) Tickets are numbered from 1 to 10. Two tickets are drawn one after the other at random. Find the probability that the number on one of the tickets is a multiple of 5 and on the other a multiple of 4. 4/45

29) The probability of student A passing an examination 2/9 and of student B passing is 5/9. Assuming the two events:

A passes, B passes as independent, find the probability of:

A) only A passing the examination. 8/81

B) only one of them passing the examination. 43/81

30) The probability of a student A passing an examination is 3/7 and of student B passing is 5/7. Assuming the two events "A passes", " B passes ", as independent, find the probability of:

A) only A passing the examination. 6/49

B) only one of them passing the examination. 26/49 

31) The probability of A, B and C solving a problem are 1/3, 2/7 and 3/8 respectively. if all the three try to solve the problem simultaneously, find the probability that exactly one of them can solve it. 25/56

32) Kamal and Monica appeared for an interview for two vacancies. The probability of Kamal's selection is 1/3 and that of Monica's selection 1/5. Find the probability that

A) both of them will be selected.

B) none of them will be selected.

C) at least one of them will be selected.

D) only one of them will be selected. 1/15,8/15,7/15,2/5

33) A certain team wins with probability 0.7, loses with probability 0.2 and ties with probability 0.1 the team plays 3 games. Find the probability that the team wins at least two of the games, but not lose. 0.49

34) A can hit a target 3 times in 6 shots, B: 2 times in 6 shots and C : 4 times in 4 shots. they fix a volley. What is the probability that atleast 2 shots hit ? 2/3

35) Arun and Tarun appeared for an interview for two vacancies. The probability of Arun's selection is 1/4 and that of Tarun's rejection is 2/3. Find the probability that atleast one of them will be selected. 1/2

36) A husband and wife appear in an interview for two vacancies for the same post. The probability of husband's selection is 1/7 and that of the wife's selection is 1/5. What is the probability that:

A) both of them will be selected?

B) only one of them will be selected ? 1/35,2/7,24/35

C) none of them will be selected ?

37) A,B and C shrt to hit a target. If A hit the target 4 times and 5 trials; B hits it three times in four trials and C hits 2 times in three trials; what is the probability that the target is hit by at least two persons? 5/6

38) Three groups of children contains three girls and one boy; two girls and two boys; 1 girl and 3 boys respectively. One child is selected at random from each group. Find the chance that the three selected comprise one girl and 2 boys. 13/32

39) Thee critics review a book. Odds in favour of the book are 5: 2, 4: 3 and 3:4 respectively for 3 critics. Find the probability that the majority are in favour of the book. 209/343

40) A speaks truth in 60% of the cases and B in 90% of the cases. In what percentage of cases are they likely 

A) contradict each other in stating same fact ? 42%

B) agree in stating the same fact ? 58%

41) A speaks truth in 75% and B in 80% of the cases. In what percentage of cases are they likely to contradict each other in narrating the same incident ? 35%

42) The odds against a husband who is 45 years old, living till he is 70 are 7 : 5 and the odds against his wife who is now 36, living till she is 61 are 5:3. Find the probability that

A) the couple will be alive 25 years hence. 5/32

B) exactly one of them will be alive 25 years hence. 23/48

C) none of them will be a alive 25 years hence,. 35/

D) atleast one of them will be alive 25 years hence. 61/96

43) A clerk was asked to mail 3 report cards to 3 students. He addresses 3 envelopes but unfortunately paid no attention to which report card be put in which envelope. what is the probability that exactly one of the student received his her own card ? 1/2

44) Neelam is taking up subjects Mathematics, Physics and Chemistry. She estimates that her probabilities of receiving grade A in these courses are 0.2, 0.3 and 0.9 respectively. If the grades can be regarded as independent events, find the Probabilities that she receives

A) All A's. 0.054

B) No A's. 0.056

C) Exactly two A's. 0.348

45) A doctor claims that 60% of the patient he examines are allergic to some type of weed. What is the probability that 

A) exactly 3 of his next 4 patients are allergic to weeds. 216/625

B) none of his next 4 patients is allergic to weeds ? 16/625 

46) Two persons A and B throw a dice alternatively till one of them gets a '3' and wins the game. Find their respectively Probabilities of winning, if A begins. 5/11

47) A and B throw alternatively a pair of dice. A wins if he throws 6 before B throws 7 and B wins if the throw 7 before A throws 6. Find their respective chance of winning, if A begins. 31/61

48) 3 person A,B,C throw a dice in succession will one gets a '6' and wins the game. Find their respective Probabilities of winning, if a begins. 25/91

49) A and B toss a Coin alternatively till one of them gets a head and wins the game find the probability that we will win the game. If A starts the game, find the probability that B will win the game. 1/3

50) In a family, the husband tells a lie in 30% cases and the wife in 35% cases. Find the probability that both contradict each other on same fact. 0.44

51) A,B, C are independent witness of an event which is a known to have occurred. A speaks the truth three times out of four, B four times out of five and C five times out of six. What is the probability that the occurrence will be reported truthfully by majority of three witnesses? 107/120

52) X is taking up subjects-- Mathematics, Physics and Chemistry in the examination. His Probabilities of getting grade A in these subjects are 0.2, 0.3 and 0.5 respectively. Find the probability that he gets

A) grade A in all subjects. 0.03

B) grade A in no subject. 0.28

C) grade A in two subjects. 0.22

53) A and B tame turns in throwing two dies, the first to throw 9 being awarded the prize. Show that their chance of winning are in the ratio 9:8.

54) A,B and C in order toss a coin. The one to throw a head wins. What are their respective chances of winning assuming that the game may continue indefinitely. 4/7,2/7,1/7

55) 3 person A,B,C throw a dice in succession till one gets a '6' and wins the game. Find their respective probabilities of winning. 36/91,30/91,25/91

56) A and B take turns in throwing two dice, the first to throw 10 being awarded the prize, show that if A has the first throw, their chance of winning are in the ratio 12:11.              

57) Fatima and John appear in an interview for two vacancies for the same post. The probability of Fatima's selection 1/7 and that of John's selection is 1/5. What is the probability that:

A) both of them will be selected.

B) only one of them will be selected ? 1/35,2/7,24/35

C) none of them will be selected ?

58) Out of 100 students, two sections of 40 and 60 are formed. If you and your friend are among 100 students, what is the probability that :

A) you both enter the same section ?

B) you both enter the different sections? 17/33, 16/33

59) In a hockey match, both teams A and B scored same number of goals upto the end of the game, so to decide the winner, the referee asked both the captains to throw a die alternatively and decide that the team, whose captain gets a first six, will be declared the winner. If the captain of Team A was asked to start, find their respective Probabilities of winning the match and state whether the decision of the refree was fair or not.         

 Team A:6/11, Team B: 5/11; The decision was fair as the two Probabilities are almost equal.

Exercise - I

Continue....

          BAYE'S THEOREM

Exercise -J

1) In a bolt factory, machines A,B,C manufacture 25%,35%,40% of the total bolts. Of their output 5,4 and 2% are defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found to be defective, what is the probability that it is manufactured by the machine B?                           28/69

2) Three Urns contains 6 red 4 black; 4 red, 6 black, and 5 red and 5 black balls. One of the urns is selected at random and a ball is drawn from it. If the ball drawn is red, find the probability that is drawn from the first urn.           2/5

3) There are 3 bags, each containing 5 white and 3 black balls. Also there are 2 bags, each containing 2 white and 4 black balls. A white ball is drawn at random. Find the probability that this white ball is from a bag of the first group.                              45/61

4) Urn-1 contains 5 red and 5 black balls, Urn-2 contains 4 red and 8 black balls and Urn-3 contains 3 red and 6 black balls. One urn is chosen at random and a ball is drawn. The colour of the ball is black. What is the probability that is has been drawn from Urn-3?                   4/11

5) A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.           3/8

6) In a test, an examinee either guesses or copies or knows the answer to a multiple choice question with four choices. The probability that he makes a guess is 1/3 and the probability that he copies the answer is 1/6. The probability that his answer is correct, given that he copied it, is 1/8. Find the probability that he knew the answer to the question, given that he correctly answered it.               24/29

7) A card from a pack of 52 Cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be hearts. Find the probability of the missing card to be a heart.                                   11/50

8) An insurance company insured 6000 scooter- drivers, 3000 car drivers and 9000 truck drivers. The probability of an accident involving a scooter, a car and a truck is 0.02, 0.06 and 0.03 respectively. One of the insured persons meets with an accident. Find the probability that he is a car driver.                 6/19 

9) A firm produces steel pipes in three plants A, B and C, with daily production of 500, 1000 and 2000 units respectively. If is known that fractions of defective output produced by three are respectively .005, .008, and.010. A pipe is selected at random from a day's total production and found to be defective. What is the probability that it came from the first plant.   5/61

10) Marksmen A and B compete by taking turns to shoot at a target. Odds in favour of A hitting the target (in a single try) are 3:2 and the odds in favour of B hitting the target (in a single try) are 4:3.

Calculate the probability of A winning the competition if he gets the first chance to shoot.        21/29

11) A letter is known to have come either from TATANAGAR or CALCUTTA. On the envelope just two consecutive letters TA are visible. What is the probability that the letter has come from

A) Calcutta.                              4/11

B) Tatanagar.                           7/11

EXERCISE - K

MATHEMATICAL EXPECTATION 

1) a) A random variable x has the following distribution, find the expectation, Mean, S. D Variance.

x:   0      1       2       3 

P: 1/8   3/8    3/8   1/8.          1.5 0.87

b) x:    2      3        4     5 

     P:  0.2    0.4    0.3  0.1      3.3,0.81

c) x:  4     6       7     10

    P: 0.2  0.4   0.3    0.1      6.3, 1.62

2) A card is drawn at random from a pack of 52 cards. if ace counts one, king, queen and jack count 10 each and other count at their face value, find the expectation of the value of the card.                    85/13

3) Find the expected number of tails in tossing three coins.      1.5

4) Find the mathematical expectation of number of tails in thowing two coins.                         1

5) 4 coins are tossed. If x denotes the number of heads, find the expected value and variance of x.           2,1

6) If a coin is tossed 100 times, in how many times will you expected head ?                                    50

7) A boy throws a coin four times and guesses each time whether the head or the tail has been thrown. He was not allowed to see the result. He is to receive ₹2 for 2 heads, ₹4 for 3 heads and ₹6 for 4 heads. Find his expectation.   2.13

8) A man draws two balls from a bag containing 3 white and 5 black balls. If he is to receive ₹14 for every white balls and ₹7 for every black balls, what is his expectation ?                  19.25

9)  A bag contains 5 white and 7 black balls. Find the expectation of a man who is allowed to draw two balls from the bag and who is receive Rs 1 for each black ball and ₹2 for each white ball drawn.     2.83

10) A box contains 8 tickets. 3 of the tickets carry a prize of ₹5 each the other 5 a prize of ₹2 each.

A) If one ticket is drawn what is the expected value of the prize ?

B) if two tickets are drawn what is the expected value of the game ?           ₹3.12, ₹6.25

11) A box contains 4 white and 6 red balls. If 2 balls are drawn from it, then find the mathematical expectation of the number of red balls.                                       1.2

12) A bag contains 2 white balls and 3 black balls. Four persons A, B, C and D in this order, draw a ball from the bag and do not replace it. The first person to draw a white ball is to receive ₹20. Determine their expectation.            ₹8,6, 4, 2

13) A box contains 4 red and 3 white balls. 2 balls are drawn from it and a man is to receive ₹20 for each white ball and lose ₹10 for each red ball. Find the amount expected by the man from the game.                                   ₹2.86

14) In a box there are 5 watches of which 2 watches are known to be defective. 2 watches are taken out at random. Let X denote the number of defective watches selected. Obtain the probability distribution of X. Also calculate E(X).         4/5

15) Find the mathematical expectation of the number of points if a balance die is rolled.   3.5

16) The probability that there is atleast one error in an account statement prepared by A is 0.3 and for B and C, they are 0.4 and 0.45 respectively. A, B and C prepared 20, 10, 40 statements respectively. Find the expected number of correct statements in all.              42

17) Find the expected value and variance of number of points in rolling two balanced dice.    7, 5.833

18) Find the expected value of the product of the points on two dice.     49/4

19) A number is chosen at random from the set 1,2,3,...100 and another number is chosen at random from the set 1, 2,3,....50. what is the expected value of the product?          1287.75

20) A and B toss in turn an ordinary die for a prize of ₹55. The first to toss a six wins. If A has the first throw what is his expectation.   ₹30

21) A and B play for a prize of ₹99. The prize is to be own by a player who first throw a  '3' with one die. A first throw and if he fails B throws and if B fails A again throws, and so on. Find their respective expectations.                         54, 45

22) The monthly demand for TV sets is known to have the following probability distribution:

Demand (x): 1    2      3       4         5

Prob(p):   0.10  0.15  0.20  0.35 0.20 The total cost (₹ y) of producing x TV sets is given by y= 100000 + 2000x. Find expected demand and expected cost.         3.40, 106800

23) A and B play for a prize of ₹152. They throw a pair of dice in succession. A win if he throws 8 before B throws 9 and B wins if he throws A throws 8. If A begins, find their respective expectations.    90, 62

Multiple choice questions 

1) One card dron drawn from a pack of 52 cards. The probability that it is the card of a king or spade is

A) 1/26 B) 3/26 C) 4/13 D) 3/13

2) Two dice are thrown together. The probability that atleast one will show its digit greater than 3 is

A) 1/4 B) 3/4 C) 1/2 D) 1/8

3) Two dice are thrown together. The probability of getting a total score of 5 is

A) 1/18 B) 1/12 C) 1/9 D) none

4) Two dice are thrown together. The probability of getting total score of seven is

A) 5/36 B) 6/36 C) 7/36 D) 8/36

5) The probability of getting a total of 10 in a single throw of two dice is

A) 1/9 B) 1/12 C) 1/6 D) 5/36

6) A card is drawn at random from a pack of 100 cards numbered 1 to 100. The probability of drawing a number which is a square is

A) 1/5 B) 2/5 C) 1/10 D) none 

7) A bag contains 3 red, 4 white and 5 blue balls. All balls are different. Two balls are drawn at random. The probability that they are of different colour is

A) 46/66 B) 10/33 C) 1/3 D) 1

8) Two dice are thrown together. The probability that neither they show equal digits nor the sum of their digits is 9 will be 

A) 13/15 B) 13/18 C) 1/9 D) 8/9

9) Four persons are selected at random out of 3 men, 2 women and 4 children. The probability that there are exactly 2 children in the selection is

A) 11/21 B) 9/21 C) 10/21 D) none 

10) The probabilities of happening of two events A and B are 0.25 and 0.50 respectively. If the probability of happening of A and B together is 0.14, then Probability that neither A nor B happens is

A) 0.39 B) 0.25 C) 0.11 D) none

11) A die is thrown, then the probability that a number 1 or 6 may occur is

A) 2/3 B) 5/6 C) 1/3 D) 1/2

12) Six boys and six girls sit in a row randomly. The probability that all girls sit together is

A) 1/122 B) 1/112 C) 1/102 D) 1/132

13) The probabilities of three mutually exclusive events A, B and C are given by 2/3, 1/4 and 1/6 respectively. The statement

A) is true B) is false C) nothing can be said D) could be either

14) (1-3p)/2, (1+ 4p)/3, (1+ p)/6 are the probabilities of three mutually exclusive and exhaustive events, then the set of all values of p is

A) (0,1) B) (-1/4,1/3) C) (0,1/3) D) n

15) A pack of cards contains 4 aces, 4 kings, 4 queens and 4 jacks. Two cards are drawn at random. The probability that atleast one of them is an ace is

A) 1/5 B) 3/16 C) 9/20 D) 1/9

16) If three dice are thrown simultaneously, then the probability of getting a score of 5 is

A) 5/216 B) 1/6 C) 1/36 D) none

17) One of the two events must occur. If the chance of one is 2/3 of the other, then odd in favour of the other are

A) 1:3 B) 3:1 C) 2:3 D) 3:2

18) The probability that a leap year will have 53 Friday or Saturday is

A) 2/7 B) 3/7 C) 4/7 D) 1/7

19) A person write 4 letters and addresses 4 envelopes. If the letter are placed in the envelopes at random, then the probability that all letters are not placed in the right envelopes, is

A) 1/4 B) 11/24 C) 15/24 D) 23/24

20) A and B are two events such that P(A)= 0.25 and P(B)= 0.50. the probability both happening together is 0.14. the probability of both A and B not happening is

A) 0.39 B) 0.25 C) 0.11 D) none

21) If the probability of A to fail in an examination is 1/5 And that of B is 3/10. Then, the probability that either A or B fails is

A) 1/2 B) 11/25 C) 19/50 D) none

22) A box contains 10 good article and 6 defective articles. One item is drawn at random. The probability that it is either good or has a defect, is

A) 64/64 B) 49/64 C) 40/64 D) 24/64

23) Three integers are selected at random from the first 20 integers. The probability that their product is even is

A) 2/19 B) 3/29 C) 17/19 D) 4/19

24) Out of 30 consecutive integers, 2 are chosen at random. The probability that their sum is odd, is

A) 14/29 B) 16/29 C) 15/29 D) 10/29

25) A bag contains 5 black balls, 4 white balls and 3 red balls. If a ball is selected randomwise, the probability that it is black or red ball is

A) 1/3 B) 1/4 C) 5/12 D) 2/3

26) Two dice are thrown simultaneously. The probability of getting a pair of aces is

A) 1/36 B) 1/3 C) 1/6 D) none

27) An urn contains 9 balls two of which are red, three blue and four black. Three balls are drawn at random. The probability that they are of the same colour is

A) 5/84 B) 3/9 C) 3/7 D) 7/17

28) Five persons entered the lift cabin on the ground floor of an 8 floor house. Suppose that each of them independently and with equal probability can leave the cabin at any floor beginning with the first, then the probability of all 5 persons leaving at different floor is

A) 7P5/7⁵ B) 7⁵/7P5 C) 6/6P5 D) 5P5/5⁵

29) A box contains 10 good article and 6 defective. One item is drawn at random. The probability that it is either good or has a defect is

A) 64/64 B) 49/64 C) 40/64 D) 24/64

30) A box contains 6 bolts and 10 nuts. Half of the bolts and half of the nuts are rusted. If one item is chosen at random, the probability that it is rusted or is a bolt is

A) 3/16 B) 5/16 C) 11/16 D) 14/16

31) If S is the sample space and P(A)= 1/3 P(B) and S= A U B, where A and B are two mutually exclusive events, then P(A) is

A) 1/4 B) 1/2 C) 3/4 D) 3/8

32) Of P(A UB)= 0.8 and P(A intersection B)= 0.3, then P(A')+ P(B') is

A) 0.3 B) 0.5 C) 0.7 D) 0.9

BINOMIAL DISTRIBUTION 

1) The probability distribution of a discrete variable x is given by:

X:        1       2      3

F(X):  2p      p     4p  find

A) E(x).                                      2.29

B) V(x).                                      0.78

C) Pr(x> 1).                                5/7

2) A random variable x has the following Probability distribution:

x:      0  1   2   3    4    5     6     7      8

P(x): k 2k 4k  6k  8k 10k12k 14k 16k

Find

A) E(x).                                         5.59

B) V(x).                                        4.27

C) p(x <3).                                 7/73

D) p(x ≥ 3).                               66/73

E) p(0< x < 5).                          20/73 

3) If a coin is tossed 3 times, obtain the p.m.f. of the number of heads. Hence obtain its probability distribution. 

4) A box contains 4 and 3 white balls. If 2 balls are drawn from this box, find the p.m.f. of the number of red balls and hence obtain its probability distribution.

5) The mean of a Binomial Distribution is 4 and the standard deviation is 3 -- This statement can not be true, why ?

6) For a Binomial Distribution, the mean is 3 and variance is 2. Find the value of n and p. Hence find the probability that X (the variable value) is 5.          9, 1/3, 224/2187

7) If the mean and the variance of a Binomial distribution are respectively 4 and 8/3, find the values of n and p.              12, 1/3 

8) The mean of a Binomial Distribution is 40 and standard deviation is 6. Calculate n, p and q.    400, 1/10, 9/10 

9) In a Binomial distribution consisting of 5 independent trials, Probabilities of 1 and 2 success are 0.4096 and 0.2048 respectively. Find the parameter p of the distribution.                       1/5

10) With usual notations, find p for a Binomial random variable X if n= 6, and if P(X= 4) = P(X= 2).      1/4

11) An experiment succeeds twice as often as it fails. What is the probability that in the next 5 trials there will be 

A) three success.            80/243

B) at least three success.    64/81

12) An experiment succeed twice as many as times as it fails. Find the chance that in 6 trials, there will be at least five success.    256/729

13) Four coins are tossed simultaneously. What is the probability of getting 2 heads and 2 tails.                                        31/81

14) Eight coins are thrown simultaneously. Find the chance of obtaining.

A) atleast 6 heads.               37/256

B) no heads.                            1/256

C) all heads.     1/256

15) If ten fair coins are tossed, what is the probability that are n of getting to tell you fire point are not more than three heads.         11/64

16) A dice thrown three times. If getting a six is considered a success, find the probability of getting atleast two success.    2/27

17) If 10 coins are tossed 100 times. How many times would you except 7 coins to fall head upward ?    12

18) What is the probability that if a fair coin is tossed 6 times we will get:

A) exactly two heads ?          15/64

B) at least 2 heads.                57/64

19) An unbiased cubic die is tossed 4 times. What is the probability of getting

A) no six.                            625/1296

B) least one six.                 671/1296

C) all odd numbers.                  1/16

D) atleast one even number.   15/16

20) If a die is thrown six times, calculate the probability that:

A) a score of 3 or less occurs on exactly 2 throws.                      15/64

B) a score of more than 2 occurs on exactly 3 throws.                160/729

C) a score of 5 or less occurs atleast once.             46655/46656

D) a score of two or less occurs on at least five occasions.     13/729

21) Assume that on an average one telephone out of 10 is busy. 6 telephone numbers are randomly selected and called. Find the probability that four of them will be busy.                                  0.001215

22) The probability that a college student will be graduate is 0.4. Determine the probability that out of 5 students

A) none.                                      0.08

B) one.                                        0.26

C)  atleast one will be graduate.   0.92

23) A machine produces 2% of  defectives on the average. If 4 articles, are chosen randomly, what is the probability that there will be exactly 2 defective articles.  0.0023

24) 25% of the inhabitants in a large town are bespectacled. What is the probability that a randomly selected group of 6 inhabitants will included Atmost 2 bespectacled persons.    1701/2048

25) The overall percentage of failures in a certain examination is 40. What is the probability that out of a group of 6 candidates atleast 4 passed the examination ?    1701/3125

26) Find the probability that in a family of 5 children there will be

A) at least one boy.       

B) at least one boy and one girl. (Assume that the probability of a female birth is 1/2).     31/32, 15/16

27) In a shooting competition, the probability of a man hitting a target is 1/5. If he fires 5 times, What is the probability of hitting the target atleast twice ?                  821/3125

28) The incidence of occupational disease in an industry is such that the workers have a 20% chance of suffering from it.  What is the probability that out of 6 workers, 4 or more will contact the disease ?     53/3125

29) In 10 independent throws of a defective die, the probability that an even number will appear 5 times is twice the probability that an even number will appear 4 times. Find the probability that an even number will not appear at all in 10 independent throws of the die.   (3/8)¹⁰

30) Suppose that half the population of a town are consumers of rice. 100 investigators are appointed to find out its truth. Each investigator interview 10 individual. How many investigators do you expect to report that three or less of the people interviewed are consumers of rice.                                          17

31) If on the average rainfalls on 12 days in every 30 days, find the probability

A) that the first four days of a given week will be fine, and the remainder wer.                   648/78125

B) that rain will fail on just 3 days of a given week.         4536/15625

32) A coin is tossed 8 times. Find the probability of getting 

A) three times head.                7/32

B) first three times head.     1/256

33) A die is tossed 5 times. Find the probability of getting face 6.

A) 3 times.                        125/3888

B) at least three times.       23/648

C) atmost three times.    3875/3888

D) first three times.            25/7776

34) A machine produces 25% defective items. From a day's production two samples of size 7 and 5 are chosen at random. Find the probability that two samples together will contain more than 25% defective items.                         0.35

35) Three fair coins are tossed 3000 times. Find the frequency distribution of number of heads,  calculate mean and standard deviations of the distribution.    1500, 27.38

        POISSON DISTRIBUTION

1) If a random variable X follows  Poisson distribution, such that P(X=1)=P(X=2), then mean of the Distribution.                2

2) If a random variable X follows Poisson distribution, such that P(X=1)=P(X=2), then mean and variance are..               2,2 

3) If a random variable X follows  Poisson distribution, such that P(X=0)=P(X=1), then P(X= 2)=?             1/2e

4) For a Poisson distribution X, if P(X=0)= 0.2, then the variance of the Distribution is:                 log 5 

5) The mean of a Poisson Distribution is 0.5, then ratio of P(X=3) to P(X=2) is ?     1:6 

6) For a Poisson variate X, P(X=2)= 3P(X=3), then the mean of X is:            1.00 

7) If X is a Poisson Variate withP(X=0)= 0.80, then variance is:                 log(5/4) 

8) A random variable X follows Poisson distribution with parameter 4. Find probability that: (given e⁻⁴=0.0183)

a) P(X=0).              0.0183 

b) P(X=1).               0.0732

c) P(X=2).               0.1464 

d) P(X=3).               0.1952 

e) P(X=4).               0.1952 

f) P(X=5).               0.15616 

g) P(X<2).              0.0915 

h) P(X is atleast 3).      0.7621 

i) P(X is almost 2).       0.2379 

j) P(X is more than 3).    0.4081 

k) P(X is 2 or more than 2).     0.2497 

l) P(3 <X<5).             0.1952 

m) P(3≤ X<5).          0.3904 

n) The standard Deviation of the given Poisson distribution is ?       2   

9) If 3% of the bolts manufactured by the company are defective. What is the probability that is a sample at 200 bolts.         (e⁻⁶= 0.00248)

a) 5 bolts will be defective ?          0.160 

b) None is defective ?                 0.00248 

10) The variance of a Poisson distribution is 4. Find the probability x=3, (e⁻⁴=0.0183).            0.1952 

11) The standard Deviation of a Poisson Variate is √3. Find the probability that x=2   (e⁻³=0.0498).         0.2241

12) The standard Deviation of a Poisson Variate is 2. Find the probability that x=2.   (e⁻⁴= 0.0183).           0.1464 

13) If the probability that an individual suffers a bad reaction from a particular injection is 0.01. find the probability that out of 500 individuals:.     (e⁻⁵= 0.00674)

a) Exactly two will suffer from bad reaction ?            0.08425  

b) More than 2 individuals will suffer a bad reaction.        0.87551

14) In a company manufacturing toys. It is found that 1 in 500 is defective. Find the probability that there will be at the most two defective in a sample of 2000 units.  (e⁻⁴= 0.0183).         0.2379 

15) If 2% of the items made by a factory are defective. Find the probability that the there are 3 defective items in a sample of 100 items.         0.180 

16) Experience has shown that, as the average, 2% of the airline's flights suffer a minor equipment failure in an aircraft. Estimate the probability that the number of minor equipment failure in the next 50 flights will be :

a) 0(zero).         0.3879 

b) atleast 2(Two).       0.4424

17) Between 4 and 5 P. M, the average number of phone calls per minute coming in to the switchboard of the company is 3. Find the probability that in one particular minute there will be:   (Given e⁻³= 0.0498) 

a) No phone call.           0.0498 

b) Exactly 2 phone calls.        0.2241

18) Between the hours 2 p.m. and 4 p.m. the average number of phone calls coming into the switch board of the company is 2.35. find the probability that in a particular minute there will be at most 2 phone calls. (e⁻²•³⁵= 0.095374) .        0.582854

19) Assume that 4% of the output of the factory making certain parts is defective and that 100 units.are in package, what is the probability that almost 3 defective parts may be found in a package. (e⁻⁴= 0.0183).             0.4331 

20) It is found that the number of accidents in a factory follows poisson distribution with a mean of two accidents for week. (e⁻²=0.135). 

a) Find the probability that no accident occurs in a week.    0.135 

b) Find the probability that the number of accident in a week exceeds 2.      0.325 

21) The number of accidents attributed in a year to the taxi drivers in a city follows poisson distribution with mean 3. Out of 1000 taxi drivers, find approximately (e⁻¹= 0.3879, e⁻²= 0.1353, e⁻³= 0.0498).

a) the number of taxi driver with no accident in a year.          50  

b) The number of taxi drivers with more than 3 accidents in a year.        353 

22) The average number of misprints per page of a book is 1.5.  What is the probability that a particular page is free from  misprints?        0.22

23) taking data from the previous questions; if the book contains 800 pages. how many of these contain more than one misprint?(e⁻¹= 0.22).      360 

24) If the chance of being killed by flood during a year is 1/3000, use Poisson distribution to calculate the probability that out of 3000 persons living in a village at least one would die in a flood in a year.        1 - e⁻¹  

25) A radioactive source emits on the averaged 2.5 particles per second. Calculate the two or more particles will be emitted in an interval of 4 seconds.      1-11e⁻¹⁰ 

26) In turning out certain toys in a manufacturing process in a factory, the average number of defective is 10%. What is the probability that exactly 3 defectives in a sample of 10 toys chosen at random, by using Poissonon approximation to the Binomial Distribution (e= 2.72).         0.061

27) 1/5% of the blades produced by a blade manufacturing factory turn out to be defective. The blades are supplied in packet of 10. Use Poisson distribution to calculate the number of packets in a consignment of 100000 packets. (11e⁻⁰•⁰²= 0.9802).

a) containing no defective.         98020 

b) containing one defective.         1960 

c) containing 2 defectives           20 

NORMAL DISTRIBUTION

 1) Find the area under the normal curve: 

a) For z= 1.54.             0.4382

b) to the right of z= 0.25.       0.4013

c) to the left of z= 1.96.       0.9750

d) lies between 0 and 1.83.     0.4664

e) is greater than 1.54.          0.0618

f) is greater than -0.86.          0.8051

g) lies between 0.43 and 1.12.       0,2022

h) is less than 0.77.           0.7794

2) Assume the mean height of soldiers to be 68.22 inches with a variance of 10.8 inches. How many soldiers in a regiment of 1000 would you expect to be over six feet tall ?     125

3) The height distribution of a group of 10000 men is normal with the mean height 64.5" and s.d 4.5". Find the number of men whose height is

a) less than 69" but greater than 55.5".      8200

b) less than 55.5".     200

c) more than 73.5".       200

4) The mean weight of 500 male students at a certain college is 151 lbs and the standard deviation is 15 lbs. Assuming that the weight are normally distributed, find how many students weight

a) between 120 and 155 lbs.    294 app

b) more than 155 lbs.       197 app

(given ∅(0.27)= 0.6064 and ∅(2.07)= 0.9808, where ∅(t) denotes the area under standard normal curve to the left the ordinate at t).

5) A sample of 100 dry battery cells tested to find the length of life produced the following result: mean =12 hours, s.d= 3 hours. Assuming that the data are normally distributed. what percentage of battery cells are expected to have life 

a) more than 15 hours.           15.87%

b) less than 6 hours.      2.28%

c) between 10 and 14 hours ?     49.74%

 given  z      2.5      2          1           0.67 

       Area  .4938  .4772  .3413     .2487

6) Assume that the mean height of soldiers in a regiment of 1000 to be 68.22" with a variance of 10.8. How many soldiers in the regiment would you expect to be over 72" tall.          125 

7) In a sample of 120 workers in a factory, the mean and standard deviation of wages were Rs11.35 and Rs3.03 respectively. Find the percentage of workers getting wages between Rs9 and Rs17 in the whole factory assuming that the wages are normally distributed.       75.1%

8) The mean yield for a one-acre plot is 662 kilos with a standard deviation of 32 kilos. Assuming normal distribution, how many 1-acre plots in a batch of 10000 plots would you expect to be have yield 

a) over 700 kilos.       1170

b)  below 650 kilos.     3520

c) What is the lowest yield of the top 1000 plots .     702.96

Continue........

Miscellaneous -1

1) A coin is tossed 6 times. What is the probability of obtaining four or more heads ?    0.344

2) Assuming that half the population is vegetarian so that the chance of an individual being a vegetarian is 1/2 and assuming that 100 investigators can take samples of 10 individual to see whether they are vegetarians , how many investigators would you expect to report that three people or less were vegetarians ?        17

3) If the probability of defective bolt is 0.1, find 

a) the mean and standard deviation for the distribution of defective bolts in a total of 500.           50, 6.71

4) The incidence of occupational disease in an industry is such that the workmen have a 20% chance of suffering from it. What is the probability that out of 6 workmen 4 or w

more will contact disease ?         53/3125

5) Suppose on an average one housing 1000 in a certain district has a fire during a year. If there are 2000 houses in that district, what is the probability that exactly 5 houses will have a fire during the year ? (e⁻²= 0.1352).     0.36

6) 10 percent of the tools produced in a certain manufacturing process turn out to be defective. Find the probability that in a sample of 10 tools chosen at random, exactly two will be defective by using

a)  binomial distribution

b) the Poisson distribution.

7) For a binomial distribution, the mean is 3 and the variance is 2. Find the values of n and p. Hence find the probability that X (the variable value) is 5.       O, 1/3, 224/2187

8) For a binomial distribution, the mean and sd are respectively 4 and √3. Calculate the probability of a getting a non zero value from this distribution.     1-0.75¹⁶

9) If a fair dice is tossed 100 times how many 'sixes' are most likely to appear ?   16

10) If the probability of defective bolt is 1/10, Find

a) mean for the distribution of defective bolts in a total of 400.   

11) A random variable x follows Poisson distribution with parameter m=2. Find the probabilities 

a) P(x=1).       0.2706

b) P(x ≤1).       0.4059

c) P(x < 1).       0.1353

d) P(x > 1).      0.5941

e) P(1≤ x ≤ 3) given e⁻²= 0.1353.      0.7216

12) The standard deviations of a Poisson distribution is 2. Find the probability that x= 3. (e⁻⁴= 0.0183).           0.1952

13) For a Poisson distribution P(x=0)= P(x=1). Find P(x > 0).      1- e⁻¹

14) If 3% of the bolts manufactured by a company are defective, what is the probability that in a sample of 200 bolts, 5 will be defective ? (1/e⁶= 0.00248).      0.16

15) The average number of misprint per page of a book is 2. Assuming Poisson distribution, what is the probability that a particular page is free from misprint ? If the book contains 1000 pages p, how many of the pages contain more than 2 misprints ?   1000(1- 5e⁻²)

16) Suppose that the number of telephone calls and operator receives from 11.00 a.m to 11.05 a.m follows a Poisson distribution with m= 3. Find

a) the probability that the operator will receive no calls in that time interval tomorrow.

b) the probability that in the next 3 day the operator will receive a total of 1 call in that time interval. (e= 2.718).      e⁻³= 0.050; 9e⁻⁹= 0.0011

17) The average number of defects per yard on a piece of cloth is 0.9. what is the probability that a one-yard piece chosen at random contains less than 2 defects? (e⁰·⁹ = 2.46).       0.772

18) if 5% of the electric bulbs manufactured by company are defective, use Poisson distribution to find the probability that in a sample of 100 bulbs

a) none is defective.        0.007

b) 5 bulbs will be defective. (e⁻⁵= 0.007).      0.182 

19) Find the probability that atmost five defective bolts will be found in a box of 200 bolts, if it is known that 2% of such bolts are expected to be defective. (You may take the distribution to be Poisson) (e⁻⁴= 0.0183).       0.78

20) The manufacturer of a certain electronic components knows that 3% of his product is defective . He sells the component in boxes of 100 and guarantees that not more than 3 in any box will be defective. What is the probability that a box will fail to meet the guarantee ? e³= 20.1.          1- 13e⁻³= 0.1913

21) In a certain factory blades are manufactured in packets of 10. There is a 0.2% probability for any blade to be defective. Using Poisson distribution calculate approximately the number of packets contaning two defective blades in a consignment of 20000 packets. e⁻⁰·²= 0.9802.       4

22) A car hire firm has cars which it hires out day by day. The number of demands for a car on each day follows Poisson distribution with mean 1.5. Calculate the proportion of days on which 

a) neither car is used.       0.2231

b) some demand is refused .  e⁻¹·⁵ = 0.2231.      0.1913

23) As a result of test on electric light bulbs, it was found that the lifetime of a particular make was distributed normally with an average life of 1000 burning hours and standard deviation of 200 hours. Out of 10,000 bulbs produced by the company how many balls are expected to fail

a) in the first 800 burning hours.       1587

b) between 800 and 1200 burning hours? Given £(1)= 0.84134.      6827

24) Assume the mean height of soldiers to be 68.22 inches with a variance of 10.8 square inches. How many soldiers in a regiment of 1000 would you expect to be over 6 ft. tall ? (Given that the area under the standard normal curve between x=0 and x= 0.35 is 0.1368 and between x=0 and x= 1.15 is 0.3749).       125

25) The mean I. Q of a group of children is 90 with a deviation of 20. Assuming that IQ is normally distributed, find the percentage of children with IQ over 100. Given £(0.5)= 0.6915, where £(x) is the cumulative distribution of standard normal distribution).      30.85

26) A die is tossed 1200 times. Find the probability that the number of 'sixes' lies between 190 and 210. (Given that the area under the standard normal curve between z= 0 and z= 0.78 is 0.2823 and between z= 0 and z= 0.81 is 0.2910).        0.5820

SAMPLING AND TEST OF SIGNIFICANCE

1) Sampling of Attributes:

The sampling distribution of the number of success, being a binomial probability model, would have as its 

mean μ = np

and as its variance = σ²= npq  or σ= √npq

Standard Error of Number of Success

Standard Error (S. E) 

1) For random samples of size n,

S. E of sample mean= σ/√n

S. E of sample proportion (p)= √(pq/n)

Where σ denotes the population standard deviations (sd) p the population proportion (p + q= 1).

If the random sample is drawn without replacement from a finite population of size N, then the above formulae are modified on multiplication by the correction factor √{(N - n)/(N -1)}

2) If mean₁ and mean₂ denote the means calculated from independent random samples of size n₁ and n₂ drawn from two populations with standard deviations σ₁ and σ₂ respectively, then

S. E of (mean₁ - mean₂)= √{σ₁/n₁ + σ₂/n₂}

In particular, if the population have the same s.d σ , then

S. E of (mean₁ - mean₂)= σ √{1/n₁ + /n₂}

3) if p₁ and p₂ denote the proportion calculatd from independent random sampls of sizes n₁ and n₂ and drawn from two populations with proportion P₁  and P₂ respectively, then

S. E of (p₁ - p₂)= √{(P₁Q₁/n₁ + P₂Q₂/n₂}

In particular , if it is assumed that the two population proportion P₁ and P₂ are equal, say P₁= P₂ = P, then

S. E of (p₁ - p₂)= √{PQ(1/n₁ + 1/n₂)}

4) For random samples from a normal population with s.d σ,

S. E of sample s.d(S)= σ/√(2n)

S. E of sample variance (S²)= σ² √(2/n)

Formulae are however approximate and used only the sample size n is large (say greater than 50)

It is used as an instrument in testing a given hypothesis. The hypothesis are generally tested at 5% level of significance. If the difference between the observe and expected result is more than 1.96 standard error (S. E), we say that the result of the experiment does not support the hypothesis at 5% level or, in other words , the difference is regarded as significant, i.e., it could not have arisen due to fluctuations of a simple sampleing. 

On the other hand, if the difference between observed and expected results is less than 1.96 S. E., it is not regard as significant, i.e., it could have arisen due to fluctuations of simple sampling, i.e., we say that the result of the experiment does not provide any evidence against the hypothesis. If the difference is more than 2.58 S. E., it is considered to be significant at 1% level. In practice, quite often a hypothesis is accepted if the differences is less than 3 S. E., because the probability of a difference greater than 3 S. E, arising by chance, is only about 3 in thousand (0.27%) as 99.73% items are covered between mean + 3σ on either side of the mean, however , it must be emphasized that the use of 3σ rule is justified only if n is large. Some people apply a criterian of 2 S. E. In order to determine whether or not the difference could have arisen due to fluctuations of sampling. However , instead of 3 S. E., or 2 S. E., it is suggested that we should use either 5% level or 1% level of significance.

( In practice 5% level is more popular)

EXERCISE - 1

1) A sample random sample of size 5 is a drawn without replacement population from a finite population consisting of 41 units. if the population standard deviation is 6.25, what is the standard error of sample mean. (use finite population correction).    2.65

2) The safety limit of a crane is known to be 32 tons. The mean weight and the standard deviation of a large number of iron rods are 0.3 ton and 0.2 tons respectively. 100 rods are lifted at a time. Find the probability of an accident .         0.1587

3) it has been found that 2% of the tools produced by a certain machine are defective. What is the probability that in a shipment of 400 such tools, 3% or more will be defective ? (probability the normal deviate lies between 0 and 1.43 is 0.4236).   0.764

4) For a population of six units, the values of the characteristics are given below:

 3, 9, 6, 5, 7, 10 

Considered all possible samples of size two from the above population and show that the mean of the sample means is exactly equal to the population population mean.

5) A population consists of the four members 3, 7, 11, 15. Considered all possible samples of size two which can be drawn with replacement from this population. Find this population. Find

a) the population mean .       9

b) the population standard deviation.       √20 

c) the mean of the sampling distribution of means.

d) the standard deviation of sampling distribution of means.

 Verify (c) and (d) directly from (a) and (b) by use of suitable formula (which you are mention).

 Solved this problem if sampling is without replacement.

Approxima

tion Confidence limits (large samples)

1) For mean μ

95% confidence limits = mean of x± 1.96(S.E of mean x)

99% confidence limits = mean of x± 2.58(S.E of mean x)

Almost sure limits = mean of x± 3(S.E of mean x)

2) For Proportion P:

95% confidence limits = p ± 1.96(S.E of p)

99% confidence limits = p ± 2.58(S.E of p)

Almost sure limits = p ± 3(S.E of p)

3) For Difference of Means (μ₁ - μ₂)

95% confidence limits = (mean₁ - mean₂) ± 1.96(S.E of mean₁ - mean₂)

99% confidence limits = (mean₁ - mean₂) ± 2.58(S.E of mean₁ - mean₂)

Almost sure limits = (mean₁ - mean₂) ± 3 (S.E of mean₁ - mean₂)

4) For Difference of Proportions(P₁ - P₂):

95% confidence limits = (p₁ - p₂) ± 1.96(S.E of p₁ - p₂)

95% confidence limits = (p₁ - p₂) ± 2.58(S.E of p₁ - p₂)

Almost sure limits = (p₁ - p₂) ± 1.96(S.E of p₁ - p₂)

z = (mean - μ)/S.E

Or

z= (observed value - Expected value)/S. E

EXERCISE - 2

1) A sample of 600 screws is taken from a large consignment and 75 are found to be defective. Estimate the percentage of defectives in the consignment and assign limits within which the percentage lies.             16.55% and 8.45%

2) A random sample of 100 ball bearing selected from a shipment of 2000 ball bearings has an average diameter of 0.354 inch with a S.D = 0.48 inch. Find 95% confidence interval for the average diameter of these 2000 ball bearings.      0.3448 to 0.3632

3) A random sample of 100 articles taken from a large batch of articles contains 5 defective particles 

a) Set up 96% confidence limits for the proportion of defective articles in the batch.       0.005 and 0.095

b) if the batch contains 2696 articles set up 95% confidence interval for the population of defective articles.       0.008 to 0.092

4) 10 Life insurance Policies in a sample of 200 taken out of 50000 were found to be insured for less than Rs5000. How many policies can be reasonably expected to be insured for less than Rs5000 in the whole lot at 95% Confidence level?      Between 1000 to 4000

5) A random sample of size 10 was drawn from a normal population with an unknown mean and a variance of 44.1 (inch)². If the observations are (in inches): 65, 71, 80, 76, 78, 82, 68, 72, 65 and 81, obtain the 95% confidence interval for the population mean.         69.7 to 77.9 inches

MISCELLANEOUS PROBLEM

1) In order to introduce some incentive for higher balance in savings accounts, a random sample of size 64 savings accounts at a bank's branch was studies to estimate the average monthly balance in savings bank accounts. The mean and standard deviation were found to be a Rs8500 and Rs 2000, respectively.

Find

a) 90%.     Rs8500±411.25

b) 95%.     Rs8500±490

iii) 99%.     Rs8500±644

confidence intervals for the population mean. 

2) One of the properties of a good quality prope paper is its burst strength . Suppose a sample of 16 specimens yields mean bursting strength of 25 units and it is known from the history of such tests that the s.d. among the specimens is 5 units . Assuming normality of test results, what are the 

a) 95%.     22.55 to 27.45

b) 98%.    22.09 to 27.91

Confidence limits for the mean bursting strength from the sample ?

3) The shopping bills of customers of a departmental store are known to follow normal distribution with mean as Rs2000 and variance as Rs25000. One day, the first 100 customers' bill are found to have an average of Rs 22000. Can the first 100 customers be regarded as a truly representative or random sample of the population of all consumers?       2000±98

4) For assessing the number of monthly transactions in credit card issued by a bank, transactions in 25 cards were analysed. The analysis revealed an average of 7.4 transactions and sample standard deviation of 2.25 transactions. Find confidence limits for the monthly number of transactions by all the credit card holders of the bank ?             6.473 to 8.327

5) if the average number of customers coming to a bank's branch is found to be 20 per hour, determine the 95% confidence limits for the number of customers in an hour.    20±3.92

6) In order to improve the quality of items produced by a production process, sample of items are inspected and number of defects in each item is recorded. One such example is the number of missing rivets on the body of a bus. if the average number of missing rivets in a sample of 4 bus bodies are found to be 25, then the 95% confidence limits for the number of missing rivets in a bus are given as.     25±4.925 

7) A company wants to determine the average time to complete a certain job. The past records show that the s.d. of the completion times for all the workers in the company has been 10 days, and there is no reason to believe that this would have changed. However, the company feels that because of the procedural changes, the mean would have changed. Determine the sample size so that the company maybe 95% confident that the sample average remains within ±2 days of the population mean.     97

8) A company believes that it holds about 35% share of the colour TV marketing in a city. The company wishes to get a precise estimate of its share within a margin of error of 2%. How large a number of households should be surveyed to get the desired estimate with a confidence of 95% ? If the cost of contacting a household is Rs20, what will be total cost of the survey ? If the budget for this survey is limited to Rs10000, what accuracy can be obtained by a survey within this budget ? 2017, Rs40340, 0.04 or 4%., ≥ 1.66, ≥ 2401

9) A new Production manager of a manufacturing unit making high intensity light bulbs, wants to improve the quality of bulbs as measured by their lives. He, therefore , decides to estimate the life of bulbs, in the existing system, and states that he would like to bring about changes in the manufacturing process unless the average life of bulbs, as evidenced by continuous burning of bulbs, is more than 650 hours. He is informed that the standard deviation of the life of bulbs is 25 hours.

A sample of 100 bulbs is selected from a week's production, and the average life of these bulbs works out to be 670 hours. Does this indicate that the quality of bulbs meets the criteria stipulated by the Production Manager ?                   The production Manager should be convinced that the average life of the bulbs being made by the existing production process is up to his level of expectance, i.e., exceeding 650 hrs.

10) A random sample of 100 students from the current year's batch gives the mean CGPA as 3.55 and variance 0.04. Can we say that this is same as the mean CGPA of the previous batch which was 3.5 ?             Current year's mean CGPA is not the same as the last year's mean CGPA.

11) It has been found from experience that the mean breaking strength of a brand of thread is 500gms, with s.d. of 40gms. From the supplies, received during the last month, a sample of 16 pieces of thread was tested which showed a mean strength of 450 gms. Can we conclude that thread supplied is inferior?       The sample indicates that the thread is inferior.

12) A telephone company's records indicate that individual customers pay on an average Rs155 per month for long distance telephone calls with standard deviation Rs45. A random sample of 40 customers' bills during a given month produce a sample mean of 160 for long distance calls. At 5% significance, can we say that the company's records indicate lesser mean more than the actual i.e., actual mean is more than 155mts ?             There is no evidence to infer that records indicate lesser mean than the actual.

13) A sample size of 10 drawn from a normal population has mean as 31, and variance as 2.25. Is it reasonable to assume that the mean of the population is 30? Assume α= 0.01.            it is assume that the population mean is 30

14) A car manufacturer claims that its new car gives a mileage of at least 10 kms per litre (kmpl) of petrol. A sample of 10 cars is taken, and their mileage recorded as follows( in kmpl):

11.2, 10.7, 11.3, 11.0, 10.8, 10.7, 10.6, 10.6, 10.7, 10.4

 Is there any statistical evidence to support the claim of the manufacturer about the mileage of its car ?          There is sufficient evidence to support the company's claim that the mileage of their car is atleast 10 kmpl.

15) The main nicotine content of a brand of cigarette is 20.0 mgs. A new process is proposed to lower the nicotine content without affecting the flavour. To test the new process, 16 cigarettes are selected at random from the week's output from the Test Plant. The sample mean nicotine content is found to be 18.5 mg. If the sd of nicotin contents is calculated to be t2 mgs, is the claim of the new process justified? Use 5% level of significance.      Sample shows the evidence that the mean nicotine contains in the cigarettes is less than 20 mgms.

16) 

EXERCISE 

1) In 324 throws of a six-faced die odd points appeared 181 times. Would you say that the die is fair.             No

2) 160 heads and 240 tails were obtained in tossing a coin 400 times. Find a 95% confidence interval for the probability of a head. Does this appear to be a true coin ?  N

3) In a sample of 500 people from a village in Rajasthan, 280 are found to be rice eaters and the rest wheat eaters. Can we assume that a both the food articles are equally popular ?              We can't assume that the food articles are equally popular.

4) In a hospital 480 female and 520 male babies were born in a week. Do these figures confirm the hypothesis that males and females are born in equal number?        It can be concluded that the male and female babies are born in equal number.

                         

χ² TEST AND GOODNESS OF FIT

1) The table given below shows the data obtained during an epidermic of Cholera:

              Attacked    Not attacked    Total 

Inaculated    31                469              500 

Not inoculated 185        1315            1500 

Total            216              1784             2000

Test the effectiveness of inoculation in preventing the attack of cholera.              The inoculation is effective in preventing the attack of cholera.

2) The following table gives the classification of 100 workers according to sex and too nature of work. Test whether nature of work is independent of the sex of the worker.

                 skilled      unskilled

Males         40             20

Females     10             30            Nature of work does not seen to be independent of the sex of the worker.

3) In an experiment on immunization of cattle from tuberculosis, the following results were obtained.

                       Affected       Not affected

Inoculated         12                 26 

Not inoculated  16                  6

Calculate χ² and discuss the effect of vaccine in controlling suspectibility to tuberculosis (5% value of χ² for 1 Degree of freedom =3.84).             Vaccine is effective in controlling suspectibility to Tuberculosis 

4) Examine by any suitable method, weather the nature of area is related to voting preference in the election for which the data are tabulated below.

Votes for

      \

Area        A       B         total

Rural      620   480   1100

Urban     380  520     900

Total     1000  1000  2000             the nature of area is related to the voting preference in the election.

5) A certain drug is claimed to be effective in curing colds. In an experiment on 150 people with colds, half of them were given the drug and half of them given sugar pills. The patients' reactions to the treatment are recorded in the following table.

         helped  Harmed  No-effect   total 

Drug  150         30           70             250 

Pills   130        40            80             250 

total   280        70          150            500 

On the basis of this data can it be concluded that there is a significant difference in the effect of the drug and pills? (for v=2, χ²₀.₀₅= 5.99).             There doesn't seem to be a significant difference in the effect of the drug and sugar pills.

6) Two investigator study the income of a group of persons by the method of sampling. Following results were obtained by them:

Inves poor mid-class  well-to-do  total 

    A    160           30           10           200

    B    140          120          40           300

 total  300          150          50           500

Show that the sampling technique of atleast one of the investigators is suspected.                        The result of the experiment does not support the hypothesis. The sampling technique of atleast one of the investigators is suspected

7) Three hundred digits were chosen at random from a set of tables. The frequency of the digits were as follows:

   Digits       Frequency

     0                 28

     1                 29

     2                 33

     3                 31

     4                 26

     5                 35

     6                 32

     7                 30

     8                 31

     9                 25

Using the χ² test assess the hypothesis with the digits were distributed in equal number in the table. (The 5% value of χ² for 9 degrees of freedom is 16.92).            The calculated value of χ² is less than the table value and hence our hypothesis that the digits were distributed in equal numbers holds good

8) The figure given below are (a) the theoretical frequencies of a distribution, and (b) the frequencies of the normal distribution having the same mean, standard deviation and the total frequency as in (a).

(a) 1   5  20  28  42  22  15   5     2

(b) 1   6  18  25  40  25  18   6     1

Apply the χ² of goodness of fit.                   The calculated value of χ² is much less than table value and hence the fit is good.

9) In an experiment on the pea breeding, the following frequencies of seeds were obtained: 360 rounds and yellow, 102 wrinkled and yellow, 109 round and green, 33 wrinkled and green; total 560. Theory predict that the frequencies should be in the proportions 9:3:3:1. Use χ² test to examine correspondence between theory and experiment.                 The calculated value of χ² is much less compared to the table value. Hence there is no reason to doubt the hypothesis. There is correspondence between theory and experiment.

10) A sample analysis of examination results of 500 students was made. It was found that 220 students had failed, 170 had secured a 3rd class, 90 were placed in second class and 20 got a first class. Are these figures commensurate with the general examination result which is the ratio 4:3:2:1 for the various categories respectively (the table of χ² for 3 d.f at 5% level of significance is 7.81)?         Since the calculated value of χ² is greater than the table value our hypothesis does not hold good.

11) A certain drug was administrated to 456 males out of a total 720 in a certain locality to taste its efficiency against typhoid. The incidence of typhoid is shown below.  Find out the effectiveness of the drug against the disease.     

                Infection   no infection    total 

Admi. 

drug              144               312          456

Without

admini

the drug       192                72           264

Total              336              384           720

                      The calculated value of χ² is much greater than the table value and hence the hypothesis is rejected. We, therefore, concluded that the drug is effective in preventing typhoid.

12) 1600 families were selected at random in a city to test the belief that high income families usually send their children to public schools and low income families often send their children to government schools. The following results were obtained:

                            School

   Income                                                total 

                  public          Government

Low            494               506               1000

High.          162               438                 600

Total           656               944              1600

Test whether income and type of schools are independent.                 The table value of χ² for 1 v= 1 as 5% level of significance 3.84. The calculated value of χ² is much greater than the table value and hence the hypothesis stands rejected. We, therefore, conclude that there is association between family income and type of tooling.

ANALYSIS OF VARIANCE

Technique in One way Classified Data

We have k independent random samples (or groups of observations), one group from each of k Normal population with means μ₁, μ₂,.......μₖ and common variance σ² (unknown). On the basis of the data, it is required to test the null hypothesis that the population means are equal.

    H₀(μ₁ = μ₂=.......= μₖ)     ..........(1)

against the alternative hypothesis H₁(all μᵢ are not equal).

        The mathematical model is

           xᵢⱼ= μᵢ + eᵢⱼ   .....................(2)

Where 

xᵢⱼ denotes the j-th observation in the i-th group ;

μᵢ    ..   mean of the i-th population, and 

eᵢⱼ ...    error due to many unspecified causes.

This is called a linear Model, since xᵢⱼ is assumed to be made up of the 'sum' of effects due to different components. It is assumed that eᵢⱼ are independent and identically distributed normal variates with mean 0 and variance σ². This model may also be written in the form 

              xᵢⱼ = μ + αᵢ + eᵢⱼ .......(3)

Where.  μ  ... denotes the general effect;

              αᵢ ... effect special to the i-th population; and

              eᵢⱼ ... error component.

Then the null hypothesis ....(1) is equivalent to stating that there is no special effect due to any population, i.e.,

             H₀ (α₁ = α₂=.......= αₖ = 0) ..........(4)

against the alternative hypothesis H₁(all αᵢ are not zero).

  Thus introduce the following notations :-

k= Number of independent samples of groups,

nᵢ = Number of observations in i-th sample,

N=  Total number of observations in all the samples = ᵢ ∑ nᵢ

xᵢⱼ = the j-th observation in the i-th group,

Tᵢ = Total of the i-th sample =   ⱼ∑xᵢⱼ 

T= Grand total of all observations = ᵢ∑ᵢ Tᵢ = ᵢ∑ ⱼ∑xᵢⱼ

mean of xᵢ = Mean of the i-th sample = Tᵢ/nᵢ

mean of x = Grand mean of all observations= T/N

     

     The expression are

Sum of Squares = SS of all sample observations about the grand mean and is called Total Sum of Squares (TSS)

Sum of the squares With in groups (SSW)

Sum of Squares due to Error (SSE).

Sum of Squares Between groups (SSB)

The degree of freedom can similarly be spli up.

Sum of Squares:  Total SS = SSW + SSB

Degree of freedom: N -1 = (N - k)+ (k -1)

 

The mean Squares calculated on dividing the sum of the Squares by corresponding degrees of freedom.

Mean Square Within groups(MSW)= SSW ÷ (N - k)

Mean Square Between groups(MSB)= SSB ÷ (k -1)

 

Assume Normal distribution in the population with a common variance(σ²),  it can be shown that the expected value of MSW is σ², i.e.,

         E(MSW)=  σ²

Irrespective of whether the population means are equal or not. But 

        E(MSB)≥ σ²

Where the the sign of equality holds when the population means are equal.

Thus when the null hypothesis H₀ is true, both the mean Squares MSB and MSW provided independent unbiased estimates of the same population variance and (σ²). Hence , we have the test statistic.

     F= MSB/MSW

which under  H₀ follows F distribution with degrees of freedom (k -1, N - k). if the observed value of the statistic equals or exceed the theoretical F value from Statistical Tables at a specified level of significance, we reject the null hypothesis and conclude that all the population means are not equal. On the other hand, if the observed value of the static is a less than the theoretical value all the population means may be considered to be equal.

   The various of the squares, degrees of freedom, mean squares corresponding to the difference sources of the variation, and also the F values(both observed and theoretical ) are shown in the form of a table known as Analysis of Variance Table. 

Steps in computation (One way Classified Data):

1) Reduce the sample observations by subtracting a suitable constant.

2) From these reduced figures, obtain 

  i) Total(Tᵢ) for each group;

  ii) grand total (T= ∑Tᵢ) ,i.e., sum of group totals Tᵢ

  iii) total of the squares of all figures ∑∑xᵢⱼ²)

3) Calculate 

   i) Correction Factor (CF)= T²/N

   ii) Total SS = ∑∑xᵢⱼ² - CF

   iii) SSB = ∑(Tᵢ²/nᵢ) - CF

    iv) SSW = Total SS - SSB

4) Write down the Degree of Freedom (D. F).

    i) D. F. for SSB = (k -1)

    ii) D. F for SSW = (N - k)

5) Calculate the Mean Squares:

     i) MSB = SSB ÷ (k -1)

    ii) MSW = SSW ÷ (N - k)

6) Obtain the observed value of F on dividing MSB by MSW:

            F= MSB ÷ MSW

7) Consult the F-tables and obtain the theoretical value of F at (say) 5% level corresponding to the degree of freedom (k -1, N - k).

8) If the observed value of F at (6) equals or exceeds the theoretical value of F at (7) reject the null hypothesis, and conclude that all population means are not equal. Otherwise, they may be taken to be equal.

EXERCISE - A

1) A random sample of 5 motor car tyres is taken from each of 3 brands manufactured by 3 companies. The lifetime of these tyres (as measured by the mileage run) is shown below. On the basis of the data, test whether the average lifetime of the three brands of tyres are equal or not.

           Lifetime of tyres ('000 miles)

Brand      A        B       C 

               35      32      34

               34      32      33

               34       31     32

               33       28      32

               34       29      33       Since the observed of F (viz. 11.2) exceed the 1% tabulated value (viz. 6.93), we reject the null hypothesis of equality of population means, and conclude that the average lifetimes of the three brands of tyres are not equal.

2) An experimenter wished to study, the effect of 4 fertilizers on the be yield of a crop.  He divided the field into 24 plots and assigned each fertilizer at random to 6 plots. Part of his calculation are shown below:

   Source    df    SS   MS   F   F5%

Fertilizer    --   2940  --     --    3.10

Witingroup --    ---     -- 

Total           --  6212 

a) Complete the above table by filling in the value marked by ---

b)  Test at the 5% level to see whether the fertilizers differ significantly.           Since the observed value of F(viz. 5.99) is larger than the 5% tabulated value (viz. 3.10), it is  significant at 5% level. We therefore conclude that the fertilizers differ significantly.

3) Each of the following sers of observations is a random sample from a normal population.

  Sets         observations 

   I       249, 242, 247, 250,252

  II       251, 256, 255, 258

  III      266, 261, 265, 264

  IV      262, 260, 263, 262, 261, 264, 262

 test whether the population means are equal (Assume that the population standard deviation are same ).                Since the observed value of F is larger than the 1% tabulated value corresponding to df.(3, 16), we reject the null hypothesis and conclude that the means of normal populations are not equal.

Technique in Two way Classified Data

We have a random sample consisting of hk observation, which are classified according to two factors -- into h classes according to factor A, and also into k classes of the sun 4 components where is the general effect if its special to the class according to factor B. There is one observation each of the hk cells corresponding to a class of factor A and simultaneously a class of factor B. These hk observations may therefore be arranged in the form a two-way table with h rows and k columns.

      The mathematical model is that any observation is made up of the sum 4 components: 

       xᵢⱼ = μ + αᵢ + βⱼ + eᵢⱼ

Where

  μ= is the general effect.

  αᵢ = effect special to the i-th class according to factor A.

   βⱼ= effect special to the j-th class according to factor B,

   eᵢⱼ = error component.

 This is called a linear Model, since any observation xᵢⱼ is supposed to be made up of the 'sum' of effects of the various components. It is assumed that eᵢⱼ are independently distributed normal variables each with mean 0 and variance σ²(unknown ).

      In the two -way classified data, the analysis of variance may be used to decide on two types of problems simultaneously, viz.,

i) whether there is any differential effect due to classification by factor A. Null hypothesis : H₀₁ = (α₁ = α₂ = ......= αₕ =0)

ii) whether there is any differential effect due to classification by factor B. Null hypothesis:  H₀₂ = (β₁ =  β₂......= βₕ =0)

Notations

h= Number of classes by factor A, (say, number of rows ).

k= Number of classes by factors B(say, number of columns)

N= hk=  total number of observations.

Tᵢ= Total of the i-th row (i= 1,2.....h)

Tⱼ = total of the j-th column (j= 1,2,....k).

T= Grand total of all observations = ∑Tᵢ = ∑T'ⱼ =∑ ∑xᵢⱼ

mean of xᵢ = Tᵢ/k = mean of the i-th row.

mean of xⱼ = Tⱼ/h = mean of j-th column.

mean= T/N = Grand mean of all observations.

Total sum of squares = (SS between factor A)+ (SS between factor B)+ (SS due to Error)

Or,

 Total SS = SSA + SSB + SSE.

The degree of freedom(d.f) for the various SS can also be split up:

          hk - 1 = (h -1)+ (k -1)+ (h -1)(k -1).

Note:

 It will be easier to calculate the d.f. as follows: since there are hk observations in all, the d.f. for Total SS is one less , viz., hk -1; since there are h classes according to factor A, the d.f. for SSA is one less, viz. (h -1); similarly the df for SSB is one less, viz., (k -1). Again just as we have SSE = Total SS - SSA - SSB, Similarly,

(df for SSE)= (df for total SS) - (df for SSA) - (df for SSB)

                    = (hk -1) - (h -1) - (k -1)

                    = (h -1)(k -1).

Mean square are now calculated as before, on dividing the Sums of Squares by the corresponding degree of freedom.

 Mean Square between classes by factor A(MSA)= SSA ÷ (h -1).

 Mean Square between classes by factor B(MSB)= SSB ÷ (k -1)

Mean Square Error        (MSE)= SSE ÷ (h -1)(k -1)

       Assuming σ², it can be shown that the expected value of MSE is σ².

                 E(MSE)= σ²

irrespective of whether the null hypothesis H₀₁ and H₀₂ are true or not. However , in general 

      E(MSA) ≥  σ² and E(MSB)≥  σ²

But E(MSA) =  σ² only if H₀₁ is true ; and E(MSB)=  σ² only if H₀₂ is true . Thus when H₀₁ is true the statistic

   F₁ = MSA/MSE

follows F distribution with the appropriate degrees of freedom. Similarly, when H₀₂  is true, the statistic.

   F₀₂ = MSB/MSE

follows F distribution with the appropriate degrees of freedom.

Steps in computation (Two-way Classified Data)

1) Reduce the observation by substracting constant from each.

2) From the reduced figures, obtain

   i) total (Tᵢ) for each group classified according to factor A, i.e., for each row.

   ii) total (T'ⱼ) for each group classified according to factor B, i.e., for each column.

   iii) Grand total (T = ∑Tᵢ = ∑T'ⱼ) for all figures.

   iv) total of the squares of all figures (∑∑ xᵢⱼ²)

3) Calculate the CF and Sums of Squares:

   i) Correction Factor(CF)= T²/N.

   ii) Total SS = ∑∑ xᵢⱼ² - CF

   iii) SSA = ∑(Tᵢ²/k) - CF

   iv) SSB = ∑(T'ⱼ²/h) - CF

   v) SSE = Total SS - SSA - SSB

NOTE:

In the calculation of various SS, we have expressions like T²/N, Tᵢ²/k, Tⱼ²/h in each of which the divisor is the number of observations included in the corresponding totals T,  Tᵢ , T'ⱼ).

4)  Write down the Degree of freedom(DF):

  i) DF for total SS = hk -1

  ii) DF for SSA = h -1

  iii) DF for SSB = k -1

  iv) DF for SSE = (DF for total SS) minus DF for SSA and SSB.

5) Calculate the mean Squares:

  i) MSA = SSA ÷ DF

  ii) MSB = SSB ÷ DF

  iii) MSE = SSE ÷ DF

6) Obtain the observed values of F on dividing MSA and MSB by MSE:

   i) F₁ = MSA ÷ MSE

   ii) F₂ = MSB ÷ MSE

7) Consult F-tables and obtain the theoretical values of F at (say) 5% level for the appropriate degrees of freedom.

Note:

Degree of freedom for F are: Divisor used for MS in numerator, followed by divisor used for MS in denominator)

8) i)  If the observed value of F₁ exceeds the theoretical value, we conclude that classification according to factor A has a differential effect on the value of the variable. That is, the means of classes by factor A are significantly different. Otherwise, there is no differential effect.

  ii) if the observed value of F₂ exceed the theoretical value, we conclude that classification according factor B has a differential effect on the values. If not, then there is no differential effect.

C. D= s √2n. t₀.₀₂₅

Or

C. D= √error (MS) × √2n . t₀.₀₂₅ 

EXERCISE - B

1) Three experimenters determine the moisture content of samples of a powder, each men taking a sample from each of 4 consignments. The results are given below.

a) Perform an analysis of variance on these data and discuss whether there is any significant difference between consignment or between experiments.

 Experimenter     Consignment 

                             I      II     III    IV

         A                 9    10     9    10 

         B                12   11     9    11 

         C                11   12    10   12

b) Also, test at 5% level which pairs of experimenters differ significantly, if any. (Given F₀.₀₅ = 5.14 for df(2,6), F₀.₀₅ = 4.76 for df(3,6), and t₀.₀₂₅ = 2.45 for 6 df).        

Miscellaneous

1) The following table shows the retail prices (Rs per kg) of a commidity in some shops selected at random in four cities:

                Cities 

  A        B        C           D

 34      29      27         34 

 37      33      29         36 

 32      30      31         38 

 33      34      28         35 

Carry out the analysis of variance to test the significance of the differences between prices of the commodity in the four cities. (given F₀.₀₅ = 3.49 for df(3,12)

2) Prepare the analysis of variance table for the following one-way classified data and comment:

                    Weight of Balls(gms)

Machine I  Machine II  Machine III

       2.0           1.8              3.0

       2.2           2.2              2.8

       1.7           2.0              3.2

Tot. 5.9          6.0              9.0

F₀.₀₅ = 5.14 for df(2,6).

3) The following table gives the yield of three varieties of wheat, each grown on four plots and allocated completely at random. Find out if the variety Differences are significant . (Tabulated value of F for df (2,9) is 8.02 at 1% level).

     Varieties  plot yields

         1       26     27     28     31 

         2       18     19     21     22 

         3       16     18     19     19

4) Random samples of electric lamps taken from 4 batches show the following data on life (in hours) of these lamps:

Batch I:     1600, 1610,1660,1700,1680

Batch II:    1590,1680,1640,1730

Batch III:   1600,1620,1570,1570,1590

Batch IV:   1580,1600,1500,1540,1550,1590

Analyse the data and test whether the average life of the lamps in the four batches are equal or not. (F₀.₀₅ = 3.24 for df(3,16).

5) To test the effect of a small percentage of rubber added to cellulose used for making toys, several batches were mixed under practically identical conditions except for the variation in the percentage of rubber. From each batch four pieces were tested for strength. The results in suitable units are:

         Percentage of Rubber

 0.0       0.1         0.2

  8         10          12

  9          9            8

  7          8            9

  9         12          11      

Examine from an analysis of variance, whether the addition of rubber has any effect on the strength. F₀.₀₅ = 4.26 for df(2,9).        

6) Twenty observations on the yield of a certain crop grown in plots of equal size are given below. The data are split into five groups of four observations each, according to the fertilizers used.

Fertilizer used.      Yield (in units)

 A(no fertilizers)   67  63  55   58

 B                            78  71  76   70

 C                            60  69  60   65

 D                            69  64  71   70

 E                            80  70  79   78

a) Make an analysis of variance of the data stating clearly the appropriate model and the necessary assumptions.

b) Test if the fertilizers B and C differ significantly at 5% level. F₀.₀₁= 4.89 for df(4,15), and t₀.₀₂₅ = 2.13 for 15 df.

7) The following data show the production of 4 workmen on three machines. Test whether the three workmen differ in respect of mean productivity: F₀.₀₅ = 6.94 for df(2,4).

Machine.            Workmen

                   A       B       C 

 1               38     33     46

 2               36     44     42

 3               49     42     39      

8) Apply the technique of analysis of variance to the following data, showing the yield of 4 varieties of a crop each from 3 blocks, and test whether the mean yield of the varities are equal or not.

Also test the equality of the block means.

Varieties               Blocks

                      I    II     III     IV 

A                   4   8      6      8

B                   5   5      7      8

C                   6   7      9      5

F₀.₀₅ = 5.143,  F₀.₀₁ = 10.925 for df (2,6); and F₀.₀₅ = 4.757 ; F₀.₀₁ = 9.779 for DF (3,6)

9) The following table gives the estimates of acreage of cultivable but not -cultivated land out of 100 acres of total land, as obtained by three investigators in each of three districts. Perform an analysis of variance to test whether there are significant differences between investigators and districts. F₀.₀₅ = 6.94 for df (2,4).

Investigator.           District

                        I       II       III     

A                    23    28     26

B                    24    25     27

C                    24    22     26

PARTIAL AND MULTIPLE CORRELATION

Partial correlation coefficient provides a measure the relationship between the dependent variable and other variables, with the effect of the rest of the variable eliminted.

 If we donate by r₁₂.₃ the coefficient of partial correlation between X₁ and X₂ keeping X₃ constant, we find that 

   r₁₂.₃ = (r₁₂ - r₁₃ r₂₃)/{√(1- r²₁₃) √(1- r²₂₃)}  

 similarly

   r ₁₃.₂ = (r₁₃ - r₁₂ r₂₃)/{√(1- r²₁₂)√(1- r²₂₃)}

 where r₁₃.₂  is the coefficient of partial correlation between X₁ and X₃  keeping X₂ constant.

  r₂₃.₁ = (r₂₃ - r₁₂ r₁₃)/{√(1- r²₁₂)√(1- r²₁₃)}

where r₂₃.₁ is the coefficient of partial correlation between X₂ and X₃ keeping X₁ constant.

 Thus for three variables X₁ , X₂ and X₃ there will be three co-efficients of partial correlation each studying the relationship between two variables when the third is held constant.

Partial Correlation Coefficient in case of Four Variables

r₁₄.₂ = (r₁₄ - r₁₂ r₂₄)/{√(1- r²₁₂)√(1- r²₂₄)}

r₁₄.₃ = (r₁₄ - r₁₃r₃₄)/{√(1- r²₁₃)√(1- r²₃₄)}

r₁₃.₄ = (r₁₃ - r₁₄ r₃₄)/{√(1- r²₁₄)√(1- r²₃₄)}

r₁₂.₄ = (r₁₂ - r₁₄ - r₂₄)/{√(1- r²₁₄)√(1- r²₂₄)}

r₂₄.₃ = (r₂₄ - r₂₃ r₃₄)/{√(1- r²₂₃)√(1- r²₃₄)}

r₃₄.₂ = (r₃₄ - r₂₃ r₂₄)/{√(1- r²₂₃)√(1- r²₂₄)}

r₂₃.₄ = (r₂₃ - r₂₄ r₃₄)/{√(1- r²₂₄)√(1- r²₃₄)}

In a similar manner, the formula for other partial correlation coefficients, i,.e.,  r₁₂.₃ , r₁₃.₂ , r₂₃.₁ , r₂₄.₁, r₃₄.₁ can also be written.

Second order Partial Correlation Coefficients

 Second order co-efficients may be obtained from first order co-efficients. In case of four variables, if r₁₂.₃₄ is the coefficient of partial correlation between X₁ and X₂ keeping X₃ and X₄ constant, then

    r₁₂.₃₄ = (r₁₂.₄ - r₁₃.₄ r₂₃.₄)/{√(1- r²₁₃.₄)√(1- r²₂₃.₄)}

Similarly ,

   r₁₃.₂₄ = (r₁₃.₄ - r₁₂.₄ r₂₃.₁)/{√(1- r²₁₂.₄)√(1- r²₂₃.₄)}

And

   r₁₄.₂₃ = (r₁₄.₃ - r₁₂.₃ r₂₄.₃)/{√(1- r²₁₂.₃)√(1- r²₂₄.₃)}

Alternative formula giving the same results are available for all three of the second order co-efficients. They are:

   r₁₂.₃₄ = (r₁₂.₃ - r₁₄.₃ r₂₄.₃)/{√(1- r²₁₄.₃)√(1- r²₂₄.₃)}

   r₁₃.₂₄ = (r₁₃.₂ - r₁₄.₂ r₃₄.₂)/{√(1- r²₁₄.₂)√(1- r²₃₄.₂)}

   r₁₄.₂₃ = (r₁₄.₂ - r₁₃.₂ r₃₄.₂)/{√(1- r²₁₃.₂)√(1- r²₃₄.₂)}

The value of a partial correlation coefficients is usually interpreted via the corresponding co-efficient of partial determination, which is merely the square of the former,

Thus if r₁₂.₃ = 0.4 then r²₁₂.₃= 0.16

Coefficient of Multiple Correlation

The coefficient of multiple linear correlation* is represent by R₁ and it is common to add subscripts designing the variables involved. Thus R₁.₂₃₄ would represent the coefficient of multiple linear correlation between X₁ on the one hand, and X₂, X₃ and X₄ on the other. The subscript of the dependent variable is always to the left of the point.

   The coefficient of multiple correlation can be expressed in terms of r₁₂ , r₁₃ and r₂₃ as follows :

 R₁.₂₃ = √{r²₁₃ + r²₁₂ - 2r₁₂r₁₃r₂₃)/(1- r²₂₃)}

 R ₂.₁₃ = √{r²₁₂+ r²₂₃ - 2r₁₂r₁₃r₂₃)/(1- r²₁₃)}

 R ₃.₁₂ = √{r²₁₃ + r²₂₃ - 2r₁₂ r₁₃r₂₃)/(1- r²₁₂)}

By squaring formula for obtaining the value of R₁.₂₃ is as follows:

     R₁.₂₃ =√{r²₁₂ + r²₁₃.₂(1- r²₁₂)}

Or

    R²₁₂.₃ = √{r²₁₂ + r²₁₃.₂(1- r²₁₂)}

Similarly R₁.₂₄= √{r²₁₂ + r²₁₄ - 2r₁₂r₁₄r₂₄)/(1- r²₂₄)}

                R₁.₂₄ = √{r²₁₂ + r²₁₄.₂(1- r²₁₂)}

And.        R₁.₃₄ = √{(r²₁₃ + r²₁₄ - 2r₁₃r₁₄r₃₄)/(1- r²₃₄)}

 Or           R₁.₃₄ = √{r²₁₃ + r²₁₄.₃(1- r²₁₃)}

To determine a multiple co-efficients with three independent variables the following formula shall be used:

         R₁.₂₃₄ = √[1- (1- r²₁₄)(1- r²₁₃.₄)(1- r²₁₂.₃₄)]

EXERCISE - A

1) In a trivariate distribution it is found that r₁₂= 0.7, r₁₃= 0.61, r₂₃ = 0.4 find r₂₃.₁ , and r₁₃.₂.             0.504, 0.628

2) On the basis of observation made on 30 cotton plants, the total correlation of yield of cotton (X₁) number of bolls, i.e., seed vessels (X₂) and height (X₃) are found to be:

    r₁₂ =0.8, r₁₃= 0.65 and r₂₃ = 0.7   

 Compute the partial correlation between yield of cotton and the number of bolls, eliminating the effect of height.            0.635

3) The following zero-order correlation coefficients are given:

   r₁₂= 0.98, r₁₃ = 0.44 and r₂₃= 0.54

  Calculate the partial correlation coefficient between first and the third variable keeping the effect of second variable constant.       -0.532

4) Is it possible to get the following from a set of experimental data:

a) r₂₃ = 0.8, r₃₁ = -0.5, r₁₂ = 0.6.             no

b) r₂₃= 0.7, r₃₁ = -0.4, r₁₂ = 0.6.            no

5) The following zero order correlation coefficients are given : r₁₂ = 0.98, r₁₃ = 0.44 and r₂₃ = 0.54.

Calculate multiple correlation coefficient treating first variables dependent and second and third variable as independent .        +0.986

6) The correlation between a general intelligence test and school achievement in a group of children from 6 to 15 years old is 0.80. The correlation between the general intelligence test and age in the same group is 0.70 and the correlation between school achievement and age is 0.60. What is the correlation between general intelligence and school achievement in children of the same age ? Comment on your result.        0.666

7) The correlation coefficient between general intelligence test and school achievement in a group of children from 8 to 14 years old is 0.80. The correlation between school achievement and age is 0.60. What is the correlation between general intelligence school achievement in children of the same age.         0.667

8) Given the following information:

 r₁₂ = 0.20, r₁₃ = 0.40, r₂₃ = 0.50, r₁₄= 0.40, r₂₄ = 0.30, r₃₄= -0.1. Find r₄₁.₂₃.         0.871

9) If r₁₂ = 0.9, r₁₃ = 0.75, r₂₃ = 0.7, find R₁.₂₃.          0.916

Regression equation 

Y= a+ b₁X₁ + b₂X₂

Where

Y= estimated value corresponding to the dependent variable.

a= Y-intercept.

X₁ and X= values of the two independent variables.

b₁ and b₂ = slopes associated with X₁ and X₂ , respectively.

Normal Equations

∑Y = na + b₁ ∑X₁ + b₂∑X₂

∑X₁Y = a∑X₁ + b₁X₁²+ b₂∑X₁X₂

∑X₂Y = a ∑X₂ + b₁∑X₁X₂ + b₂∑X₂²

EXERCISE - B

1) 

Month         X₁        X₂         Y

January     45        16        29 

 February   42        14        24

 March       44        15        27

 April          45        13        25

May            43        13        26

June           46        14        28

July             44        16        30

August        45        16        28

September 44        15        28

October       43        15        27

Calculate the multiple regression plane.           Y= 13.828+ 0.564X₁ + 1.099X₂

2) Given the following set of data

a) calculate the multiple regression plane

b) Predict Y when X₁ =3.0 and X₂ = 2.7

Y:   25     30     11     22    27     19

X₁: 3.5    6.7    1.5    0.3   4.6    2.0

X₂: 5.0    4.2    8.5    1.4   3.6    1.3         Y= 20.3916 + 2.3403X₁ - 1.3283X₂, 23.83

3) The following information has been gather from a random sample of apartment renters in a city. We are trying to predict rent (in dollars per month) based on the size of apartment (number of a rooms) and the distance from downtown (in miles).

Rent:     360 1000 450 525 350 300

N.R :      2         6      3      4    2      1

DfD:       1         1      2      3  10     4 

a) calculate the least squares equation that the best relates these three variables.

b) If someone is looking for a two -bedroom apartment 2 miles from downtown, what rent should he expect to pay.      Y= 96.4581+ 136.4847X₁ - 2.4035X₂,  365

4) Give the following set of data

a) Calculate the multiple regression plane

b) Predict Y when X₁ = 10.5 and X₂ = 13.6

Y:  11.4  16.6  20.5  29.4  7.6   13.8  28.5

X₁: 4.5     8.7   12.6  19.7  2.9    6.7   17.4

X₂: 13.2  18.7  19.8  25.4  22.8 17.8 14.6

5) For the following set of data:

Calculate the multiple regression plane.

a) Predict Y for X₁ = 28 and X₂ = 10

Y:   10  17  18   26   35   8

X₁ :  8   21  21  17    36   9

X₂ :  4    9   11  20    13  28 

6) Given the following set of data 

a) calculate the multiple regression plane.

b)  predict Y when Y= X₁ = -1 and X₂ = 4

Y:  6   10    9   14    7      5 

X₁: 1    3     2  - 2     3      6 

X₂ :3  - 1     4    7     2    - 4

∑