PROBABILITY

1. Three six faced dice are tossed together, then the probability that exactly two of the three numbers

are equal is

1) 165/216 2) 177/216 3) 51/216 4) 90/216

2. If three six faced fair dice are thrown together, then the probability that the sum of the numbers

appearing on the dice is k(3 k 8) is

1) 432

)2k)(1k( 2)

432

)1k(k

3) 432

k2

4) 432

)2k(k

3. If three six faced fair dice are thrown together, then the probability that the sum of the numbers

appearing on the dice is k (9 k 14) is

1) 216

83k21k2 2)

216

83k21k2

3) 216

83kk21 2 4)

216

83kk21 2

4. Six faces of a die are marked with numbers 1,

–1, 0, –2, 2, 3 and the die is thrown thrice. The probability that the sum of the numbers thrown is six,

is

1) 216

3 2)

216

6 3)

216

10 4)

216

18

5. Four tickets marked 00, 01, 10, 11 respectively are placed in a bag. A ticket is drawn at random five

times, being replaced each time. The probability that the sum of the numbers on the tickets is 22 is

1) 2/7 2) 25/256 3) 132/256 4) 0

6. Four persons entered the lift cabin on the ground floor of a 7 floor house. Suppose that each of them

independently and with equal probability can leave the cabin at any floor beginning with the first. The

probability of all 4 persons leaving at different floors is

1) 5/18 2) 7/18 3) 6/18 4) 4/18

7. S is a sample space. S = {x N : 1 < x 100} and E = {x : (x + 1)(x – 1) S}. then P(E) =

1) 1/10 2) 2/25 3) 99/100 4) 1/11

8. A has 3 shares in a lottery containing 3 prizes and 6 blanks. B has two shares in a lottery containing

2 prizes and 6 blanks. The ratio of their chances of success is

1) 952 : 715 2) 274 : 659 3) 113 : 907 4) 64 : 39

9. Five horses are in a race. Mr. A selects two of the horses at random and bets on them. The

probability that Mr. A selected the winning horse is

1) 3/5 2) 1/5 3) 2/5 4) 4/5

10. Two numbers ‘a’ and ‘b’ are chosen at random from the set of first 30 natural numbers. The

probability that a2 – b

2 is divisible by 3 is

1) 9/87 2) 12/87 3) 15/87 4) 47/87

11. n persons sit in a row. The probability that 2 particular persons, never sit together is

1) !n

)!2n(C2)1n(

2) !n

)!2n(C2)1n(

3) !n

)!2n(C2)1n(

4) n

2n

12. Five digit numbers are formed with 0, 1, 2, 3, 4 (not allowing a digit being repeated in any number).

The probability of getting 2 in the ten’s place and 0 in the units place always is

1) 1/16 2) 1/32 3) 1/64 4) 1/96

13. Two integers x and y are chosen with replacement out of the set {0, 1, 2, 3, ………, 10}. Then the

probability that |x – y| > 5 is

PROBABILITY

1) 81/21 2) 30/121 3) 25/121 4) 20/121

14. 100 tickets are numbered as 00, 01, 02, …, 09, 10, 11, …, 99. When a ticket is drawn at random

from them and if A is the event of getting 9 as the sum of the numbers on the ticket, then P(A) =

1) 9/100 2) 1/10 3) 11/100 4) 3/25

15. Each of the two digit numbers that can be formed from the integers 1 to 9, with no digit repeated, is

written on a card. Then all the cards are thoroughly shuffled and stacked. If one card is drawn from

the stack, the probability that the sum of the digits in the number on it will be 10 is

1) 1/9 2) 10/ 9C2 3) 1/10 4) 1/12

16. Three integers are chosen at random without replacement from the first 20 integers. The probability

that the product is odd is

1) 17/19 2) 2/19 3) 10

C2/20

C2 4) 10

C2

17. The numbers 1, 2, 3, …, n are arranged in a random order. The probability that the digits 1,2,3,…, k

(k < n) appears as neighbours is

1) 1n

)!kn( 2)

knC

1kn 3)

knC

kn 4)

!n

!k

18. A bag contains 50 tickets numbered 1, 2, 3, …, 50 of which five are drawn at random and arranged

in ascending order of magnitude (x1 < x2 < x3 < x4 < x5). The probability that x3 = 30 is

1) 5

50

220

C

C 2)

550

229

C

C 3)

550

229

220

C

CC 4) 2

29 C

19. Twelve persons are attending a dinner party. Chairs are arranged for them to be seated in a circular

way. Of the persons attending, only 2 are ladies. The probability of having always 3 men in between

the two ladies is

1) 11

2 2)

11

1 3)

!11

2!3C310

4) 2

11

20. There are 10 stations between two cities A and B. A train is to stop at three of these 10 stations. The

probability that no two of these three stations are consecutive is

1) 7/15 2) 7/12 3) 7/10 4) 5/7

21. Two friends A and B have equal number of daughters. There are three cinema tickets which are to

be distributed among the daughters of A and B. The probability that all the tickets go to daughters of

A is 1/20. The number of daughters each of them have is

1) 4 2) 5 3) 6 4) 3

22. If a number x is selected from natural numbers 1 to 100, then the probability for x + 100/x > 29 is

1) 41/50 2) 47/50 3) 39/50 4) 37/50

23. The probability that the roots of the equation 2

n

2

1nxx2 = 0 are real, where n N such that n

5, is

1) 1/5 2) 2/5 3) 3/5 4) 4/5

24. If p is chosen at random in the interval 0 p 5, the probability that the roots of the equation

2

1

4

ppxx2 = 0 are real, is

1) 1/5 2) 2/5 3) 3/5 4) 4/5

25. The Unknown coefficient of the equation x2 + bx + 3 = 0 is determined by throwing an ordinary six

faced die. The probability that the equation has real roots is

1) 1/36 2) 4/36 3) 1/2 4) 2/3

26. There are 20 pairs of shoes in a closet. Four shoes are selected at random. The probability that

there is exactly one pair is

1) 4

40

119

120

C

CC 2)

440

238

120

C

CC

PROBABILITY

3)

440

119

238

120

C

CCC 4)

440

12

120

C

CC

27. If three squares are chosen at random on a chess board, then the chance that they should be in a

diagonal line is

1) 17/744 2) 31/744 3) 7/744 4) 1/744

28. Two squares are chosen at random on a chess board. The probability that they have a side in

common is

1) 1/9 2) 2/7 3) 1/18 4) 1/12

29. Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle

with these three vertices is equilateral is

1) 1/2 2) 1/5 3) 1/10 4) 1/20

30. 2n boys are randomly divided into two subgroups containing ‘n’ boys each. The probability that the

two tallest boys are in different groups is

1) 1n2

n

2)

1n2

1n

3)

2n4

1n2 4)

2n2

1n

31. A box contains 2 fifty-paise coins, 5 twentyfive-paise coins and a certain number n ( 2) of ten and

five paise coins. Five coins are taken out of the box at random. The probability that the total value of

these 5 coins is less than one rupee and fifty paise is

1) 5

)7n( C

)2n(10

2)

5)7n( C

)2n(101

3) 5

)2n( C

)2n(5

4) 1 –

5)2n( C

)2n(5

32. If ‘m’ rupee coins and ‘n’ ten paise coins are placed in a line, then the probability that the extreme

coins are ten paise coins is

1) mnm C 2)

)1nm)(nm(

)1n(n

3) mnm P 4) n

nm C

33. Let S be a set containing ‘n’ elements. If two subsets A and B of S are picked at random from the set

of all subsets of S, then the probability that A and B have the same number of elements is

1) n2

nn2

2

C 2)

nn2 C

1

3) n2

)1n2....(5.3.1 4)

n

n

4

3

34. A is a set containing ‘n’ elements. A subset P of A is chosen at random. The set A is reconstructed

by replacing the elements of the subset P. A subset Q of A is chosen at random. The probability that

P and Q have no common elements is

1) n)2/1( 2) n)3/2( 3) n)4/3( 4) n)5/4(

35. Let S be a set containing ‘n’ elements. Two subsets A and B of S are chosen at random. The

probability that A B = S is

1) n2

nn2

2

C 2)

n

4

3

3)

nn2 C

1 4)

n

3

4

36. A mapping is selected at random from the set of all mappings from the set A = {1, 2, 3} into B = {1, 2,

3, 4}. The probability that the mapping selected is many-to-one, is

1) 5/8 2) 3/8 3) 1/4 4) 24/64

37. A binary operation is chosen at random from the set of all binary operations on a set A containing n

elements. The probability that the binary operation is commutative is

PROBABILITY

1) 2n

n

n

n 2)

2n

2/n

n

n 3)

2/n

2/n

2

n

n 4)

n

n

n

n2

38. If three dice are thrown, the probability that they show the numbers in A.P. is

1) 1/36 2) 1/18 3) 2/9 4) 5/18

39. Out of 2n tickets numbered 1, 2, …2n; 3 are chosen at random. The probability that the numbers on

them are in A.P. is

1) 1n2

2

2)

)1n2(3

2

3)

1n2

3

4)

)1n2(2

3

40. The probability that atleast one of the events A and B occurs is 0.7 and they occur simultaneously

with probability 0.2. Then )B(P)A(P =

1) 1.8 2) 0.6 3) 1.1 4) 1.4

41. A random variable X has the probability distribution:

X : 1 2 3 4 5 6 7 8

p(X) : 0.15 0.23 0.12 0.10 0.20 0.08 0.07 0.05

For the events E = {X is a prime number} and F = {X < 4}, the probability P(E F) is

1) 0.87 2) 0.50 3) 0.35 4) 0.77

42. Three groups of children contain 3 girls and one boy; 2 girls and 2 boys; one girl and 3 boys. One

child is selected at random from each group. The probability that the three selected consist of 1 girl

and 2 boys is

1) 13/32 2) 19/32 3) 13/19 4) 6/19

43. Let A, B, C be three events such that P(A) = 0.3, P(B) = 0.4, P(C) = 0.8, P(A B)= 0.18, P(A C) =

0.28, P(A B C)= 0.09. If P(ABC) 0.75, then

1) 0.13 P(B C) 0.38

2) 0.23 P(B C) 0.75

3) 0.48 P(B C) 0.75

4) 0.23 P(B C) 0.48

44. If 2

p21,

4

p1,

3

p31 are the probabilities of three mutually exclusive events, then the set of all

values of p is

1) [1/4, 1/3] 2) [1/3, 1/2]

3) [1/4, 1/2] 4) [1/3, 2/3]

45. E1, E2 are events of a sample space such that P(E1) = 1/4, P(E2 | E1) = 1/2, P(E1 | E2) = 1/4. Then

P(E1 | E2) + P(E1 | 2E ) =

1) 1/4 2) 1/3 3) 1/2 4) 3/4

46. There are 20 cards. 10 of these cards have the letter ‘I’ printed on them and the other 10 have the

letter ‘T’ printed on them. If three cards are picked up at random and kept in the same order, the

probability of making the word IIT is

1) 4/27 2) 5/38 3) 1/8 4) 9/80

47. Three numbers are selected at random without replacement from the set of numbers {1, 2, …, n}.

The conditional probability that the third number lies between the first two, if the first number is

known to be smaller than the second, is

1) 1/6 2) 1/3 3) 1/2 4) 1/4

48. A box contains 100 tickets, numbered 1, 2,…,100. Two tickets are chosen at random. It is given that

the maximum number on the two chosen tickets is not more than 10. The minimum number of them

is 5 with probability

1) 11/15 2) 13/15 3) 13/17 4) 13/19

49. For a biased die the probabilities for different faces to turn up are given below

Face 1 2 3 4 5 6

Probability 0.1 0.32 0.21 0.15 0.05 0.17

PROBABILITY

The die is tossed and you are told that either face 1 or 2 has turned up. Then the probability that it is

face 1 is

1) 5/21 2) 6/23 3) 5/23 4) 6/21

50. One ticket is selected at random from 100 tickets numbered 00, 01, 02, …99. Suppose A and B are

the sum and product of the digits found on the ticket. Then P(A = 7 | B = 0) is given by

1) 2/13 2) 2/19 3) 1/50 4) 1/19

51. A and B are two candidates seeking admission in IIT. The probability that A is selected is 0.5 and the

probability that both A and B are selected is at most 0.3. Then

1) 0 P(B) 0.3 2) 0 P(B) 0.6

3) 0.3 P(B) 0.9 4) 0.6 P(B) 0.9

52. If the probability for A to fail in one exam is 0.2 and that for B is 0.3, then the probability that either A

or B fails is

1) 0.14 2) 0.6 3) 0.44 4) 0.24

53. The probability that a man will live 10 more years is 1/4 and the probability that his wife will live 10

more years is 1/3. Then the probability that neither will be alive in 10 more years is

1) 5/12 2) 1/2 3) 7/12 4) 11/12

54. A and B are independent events. The probability that both A and B occur is 1/6 and the probability

that neither of them occurs is 1/3. The probability of occurrence of A is

1) 1/3 2) 1/5 3) 1/2 or 1/3 4) 2/5

55. Three faces of a fair die are yellow, two faces red and one blue. The die is tossed 3 times. The

probability that the colours, yellow, red and blue appear in the first, second and the third tosses

respectively is

1) 1/2 2) 1/36 3) 6/36 4) 5/36

56. If four whole numbers taken at random are multiplied together, then the chance that the last digit in

the product is 1 or 3 or 7 or 9 is

1) 16/625 2) 609/625 3) 323/625 4) 13/625

57. If ‘n’ positive integers are taken at random and multiplied together, the probability that the last digit of

the product is 2, 4, 6 or 8 is

1) n

nn

5

35 2)

n

nn

5

24 3)

n

nn

5

23 4)

n

nn

4

23

58. The probability that an event A happens in one trial of an experiment is 0.4. Three independent trials

of the experiment are performed. The probability that the event A happens atleast once is

1) 0.936 2) 0.784 3) 0.904 4) 0.748

59. Numbers are selected at random one at a time from the two digit numbers 00, 01, 02, 03…,99 with

replacement. An event E occurs if the product of the 2 digits of a selected number is 18. If four

numbers are selected, the probability that the event E occurs atleast 3 times is

1) 94/254

2) 95/254

3) 96/254

4) 97/254

60. Fifteen coupons are numbered 1, 2, 3, …, 15 respectively. Seven coupons are selected at random

one at a time with replacement. The probability that the largest number appearing on a selected

coupon is atmost 9, is

1) (9/16)6

2) (8/15)7

3) (3/5)7

4) (9/14)7

61. Two persons each make a single throw with a die. The probability that they get equal value is p1.

Four persons each make a single throw and probability of three being equal is p2. Then

1) p1 = p2 2) p1 < p2 3) p1 > p2 4) p1/p2 = 0

62. An anti-aircraft gun can take a maximum of three shots at an enemy plane moving away from it. The

probabilities of hitting the plane at the first, second and third shot are 0.5, 0.4, 0.3 respectively. The

probability that the gun hits the plane is

1) 0.79 2) 0.488 3) 0.6976 4) 0.784

PROBABILITY

63. India plays two matches each with West Indies and Australia. In any match the probabilities of India

getting points 0,1 and 2 are 0.45, 0.05 and 0.50 respectively. Assuming that the outcomes are

independent, the probability of getting atleast 7 points is

1) 0.8750 2) 0.0875 3) 0.0625 4) 0.0250

64. In a multiple choice question there are four alternative answers, of which one or more than one is

correct. A candidate will get marks on the question only if he ticks all the correct answers. The

candidate decides to tick answers at random. If he is allowed up to three chances to answer the

question, the probability that he will get marks on it is

1) 1/3 2) 1/5 3) 3/5 4) 1/15

65. The probability that A speaks truth is 4/5, while this probability for B is 3/4. The probability that they

contradict each other when asked to speak on a fact is

1) 3/20 2) 4/5 3) 7/20 4) 1/5

66. Two persons A and B toss a die. The person who first throws 6 wins. If A starts, then the probability

of his winning is

1) 1/2 2) 5/11 3) 6/11 4) 10/11

67. A and B are rolling 2 dice, on the condition that the person who gets a sum of 9 on both the dice, will

win the game. If A starts the game, then their probabilities of winning the game are

1) 3/11, 8/11 2) 7/11, 4/11

3) 6/11, 5/11 4) 9/17, 8/17

68. A, B, C are tossing a coin on the condition that, the person who gets a head first, wins the game. If A

starts the game, then the probabilities of A, B and C to win the game are

1) 4/7, 2/7, 1/7 2) 2/5, 1/5, 2/5

3) 2/7, 3/7, 2/7 4) 1/5, 3/5, 1/5

69. A bag contains ‘a’ white and ‘b’ black balls. Two players A and B alternatively draw a ball from the

bag, replacing the ball each time after the draw till one of them draws a white ball and wins the

game. A begins the game. If the probability of A winning the game is three times that of B, then a : b

=

1) 1 : 1 2) 1 : 2 3) 2 : 1 4) 2 : 3

70. A fair die is thrown until a score of less than five points is obtained. The probability of obtaining not

less than two points on the last throw is

1) 3/4 2) 4/5 3) 5/6 4) 1/3

71. On a toss of two dice, A throws a total of 5. Then the probability that he will throw another 5 before

he throws 7 is

1) 1/9 2) 1/6 3) 2/5 4) 5/36

72. A and B throw a pair of dice. A wins if he throws 6 before B throws 7 and B wins if he throws 7 before

A throws 6. If A begins, his chance of winning is

1) 5/61 2) 30/61 3) 35/61 4) 60/61

73. A man alternately tosses a coin and throws a die beginning with the coin. The probability that he gets

a head in the coin before he gets a 5 or 6 on the die is

1) 3/4 2) 1/2 3) 1/3 4) 1/4

74. Bag A contains 3 white and 2 black balls. Bag B contains 2 white and 4 black balls. One bag is

selected at random and a ball is drawn from it. The probability that it is white is

1) 52/77 2) 76/155 3) 89/198 4) 7/15

75. Bag A contains 8 black and 5 white balls. Bag B contains 6 black and 7 white balls. A die is rolled. If

2 or 5 turns up, then choose bag A otherwise choose bag B. If one ball is drawn from the selected

bag, the probability that it is black is

1) 20/39 2) 43/90 3) 34/89 4) 52/77

76. There are two groups of subjects one of which consists of 5 Science and 3 Engineering subjects and

the other consists of 3 Science and 5 Engineering subjects. An unbiased die is cast. If 3 or 5 turns

up, a subject is selected at random from the first group, otherwise the subject is selected at random

from the second group. The probability that an Engineering subject is selected ultimately is

PROBABILITY

1) 13/16 2) 13/24 3) 13/36 4) 7/16

77. Three bags contain 3 red, 4 black balls; 4 red, 5 black balls, 5 red, 2 black balls. If one bag is

selected at random and a ball is drawn from it, then the probability that it is red is

1) 100/189 2) 25/36 3) 78/98 4) 12/55

78. A lot contains 20 articles. The probability that the lot contains exactly two defective articles is 0.4 and

the probability that it contains exactly 3 defective articles is 0.6. Articles are drawn from the lot at

random one by one, without replacement and tested till all defective articles are found. The

probability that the testing procedure ends at the 12th

testing is

1) 1900

11 2)

1900

44 3)

1900

66 4)

1900

99

79. Urn A contains 6 red and 4 black balls and urn B contains 4 red and 6 black balls. One ball is drawn

at random from urn A and placed in urn B. Then one ball is drawn at random from urn B and placed

in urn A. If one ball is now drawn from urn A, the probability that it is found to be red is

1) 32/55 2) 33/55 3) 32/63 4) 25/66

80. A bag contains 5 red, 3 black balls and another bag contains 4 red and 5 black balls. One of the

bags is chosen at random and a draw of two balls is made from it. The chance that one is red and

the other is black is

1) 71/126 2) 275/504 3) 145/345 4) 87/99

81. A bag contains 16 coins of which two are counterfeit with heads on both sides. The rest are fair

coins. One is selected at random from the bag and tossed. The probability of getting a head is

1) 9/16 2) 11/16 3) 5/9 4) 13/16

82. A bag contains four balls. Two balls are drawn and found them to be white. The probability that all

the balls are white is

1) 1/2 2) 3/5 3) 1/4 4) 4/6

83. A bag contains 6 white and 4 black balls. A fair die is rolled and a number of balls equal to that

appearing on the die is chosen from the urn at random. The probability that all the balls selected are

white is

1) 1/5 2) 1/6 3) 1/7 4) 1/8

84. A letter is known to have come either from LONDON or CLIFTON; on the postmark only the two

consecutive letters ON are legible. The probability that it came from London is

1) 4 2) 1 3) 3 4) 3 5) 2 6) 1 7) 4 8) 4 9) 3 10) 4

11) 4 12) 1 13) 2 14) 2 15) 1 16) 2 17) 2 18) 3 19) 1 20) 1

21) 4 22) 3 23) 3 24) 3 25) 3 26) 3 27) 3 28) 3 29) 3 30) 1

31) 2 32) 2 33) 1 34) 3 35) 2 36) 1 37) 3 38) 2 39) 4 40) 3

41) 4 42) 1 43) 1 44) 2 45) 3 46) 2 47) 2 48) 2 49) 1 50) 2

51) 2 52) 3 53) 2 54) 3 55) 3 56) 1 57) 1 58) 2 59) 4 60) 3

61) 3 62) 1 63) 2 64) 2 65) 3 66) 3 67) 4 68) 1 69) 3 70) 1

71) 3 72) 2 73) 1 74) 4 75) 1 76) 2 77) 1 78) 4 79) 1 80) 2

81) 1 82) 2 83) 1 84) 2 85) 3 86) 3 87) 1

PROBABILITY

1) 5/17 2) 12/17 3) 17/30 4) 3/5

85. Three boxes numbered, I, II, III contain balls as follows

White Black Red

I 1 2 3

II 2 1 1

III 4 5 3

One box is randomly selected and a ball is drawn from it. If the ball is red, then the probability that it

is from box II, is

1) 1/2 2) 1/3 3) 1/4 4) 1/9

86. A bag A contains 2 white and 3 red balls and a bag B contains 4 white and 5 red balls. One ball is

drawn at random from one of the bags and is found to be red. The probability that it was drawn from

bag B is

1) 23/54 2) 25/51 3) 25/52 4) 27/55

87. A man is known to speak the truth 3 out of 4 times. He throws a die and reports that it is a six. The

probability that it is actually a six is

1) 3/8 2) 2/7 3) 1/9 4) 4

KEY

88) 4 89) 1 90) 3 91) 3 92) 2 93) 1 94) 4 95) 4 96) 3 97) 4

98) 4 99) 1 100) 2 101) 2 102) 1 103) 2 104) 2 105) 3 106) 1 107) 1

108) 4 109) 3 110) 3 111) 3 112) 3 113) 3 114) 3 115) 3 116) 3 117) 1

118) 2 119) 2 120) 1 121) 3 122) 2 123) 1 124) 3 125) 2 126) 4 127) 3

128) 4 129) 1 130) 1 131) 2 132) 3 133) 2 134) 2 135) 2 136) 1 137) 2

138) 2 139) 3 140) 2 141) 3 142) 3 143) 1 144) 1 145) 2 146) 4 147) 3

148) 3 149) 1 150) 2 151) 2 152) 3 153) 3 154) 4 155) 1 156) 3 157) 1

158) 3 159) 2 160) 1 161) 4 162) 1 163) 2 164) 1 165) 4 166) 1 167) 2

168) 1 169) 2 170) 1 171) 2 172) 3 173) 3 174) 1