300+ TOP Probability MCQs and Answers Online Quiz Test

Probability Multiple Choice Questions

1. What is the probability that a leap year selected at random will contain 53 Sundays?

(A) 2/7

(B) 3/7

(C) 4/7

(D) 5/7

2. A fair die is rolled. Find the probability of getting a prime factor of 6.

(A) 1/3

(B) 1/2

(C) 2/3

(D) 5/6

3. Find the probability of throwing a sum 9 with two dice

(A) 1/9

(B) 1/13

(C) 1/12

(D) 1/36

4. What is the probability of getting more than 3 when a dice is thrown?

(A) 1/3

(B) 1/6

(C) 1/2

(D) 2/3

5. Two dice are thrown simultaneously then number of chances of getting sum 8 is

(A) 3

(B) 4

(C) 5

(D) 6

6. In a simultaneous throw of two dice, what is the probability of getting a total of 7?

(A) 1/6

(B) 1/3

(C) 2/3

(D) 3/4

7. In a single throw of a die, the probability of getting a multiple of 3 is

(A) 1/2

(B) 1/3

(C) 1/4

(D) 1/5

8. In a well shuffled pack of 52 cards, a card is drawn at random, find the probability of diamond or kind card

(A) 4/13

(B) 5/13

(C) 6/13

(D) 7/13

9. In a simultaneous throw of two dice, what is the probability of getting a total of 10 or 11?

(A) 1/3

(B) 1/7

(C) 5/12

(D) 5/36

10. What is probability of getting more than 5 when a dice is thrown?

(A) 1/3

(B) 1/4

(C) 1/5

(D) 1/6

11. A box contains 10 cards numbered from 1 to 10. One card is drawn from the box. Find the probability that the number on the chosen card is a multiple of 2.

(A) 2/3

(B) 1/3

(C) 1/6

(D) 1/2

12. A dice is thrown once. Find the probability of getting an even number.

(A) 3/2

(B) 1/6

(C) 1

(D) 1/2

13. Find the probability of getting a number less than 5 in a single throw of dice.

(A) 1/5

(B) 5/6

(C) 2/3

(D) 6/5

14. A die is rolled. Find the probability of getting an odd number.

(A) 1/6

(B) 1/3

(C) 1/2

(D) 1/4

15. In a bag, there are 8 red balls, 7 blue balls, and 6 green balls. One ball is picked up randomly. What is the probability that it is neither red nor green?

(A) 3/7

(B) 2/7

(C) 2/3

(D) 1/3

16. A single die is rolled. Find the probability that the number obtained is greater than 2.

(A) 2/3

(B) 1/3

(C) 1/6

(D) 1/2

17. The probability that there will be a storm tomorrow is 0.45. What is the probability that there will not be a storm tomorrow?

(A) 1/0.45

(B) 1/0.55

(C) 0.55

(D) 1

18. Cards marked with the numbers 2 to 101 are placed in a box and mixed thoroughly. One card is drawn from the box. Find the probability of a number that is a perfect cube.

(A) 3/99

(B) 3/101

(C) 3/100

(D) 1/100

19. The probability that there will be a test at school tomorrow is 0.85. What is the probability that there will NOT be a test tomorrow?

(A) 1/0.85

(B) 1/0.15

(C) 0.15

(D) 0.85

22. Let X be a radom variable following normal distribution with mean +1 and variance 4. Let Y be another normal variable with mean -1 and variance unknown. If P(X ≤ -1) = P(Y ≥ 2) the standard deviation of Y is

1. 3
2. 2
3. 1.414
4. 1

Answer: a

23. If probability density function of a random variable x is

f(x) = x² for -1 ≤ x ≤ 1, and

f(x) = 0 for any other value of x

them, the percentage probability

[Image: P\left ( \frac{-1}{3}\leq x\leq \frac{1}{3} \right )]

is

1. 0.247
2. 2.47
3. 24.7
4. 247

Answer: b

24. Let X and Y be two independent random variables. Which one of the relations between expectation (E), variance (Var) and covariance (Cov) given below is FALSE?

1. E(XY) = E(X) E(Y)
2. Cov (X, Y) = 0
3. Var (X+ Y) = Var (X) + Var (Y)
4. E(X² Y²) = (E(X))² (E(Y))²

Answer: d

25. A probability density function is of the form p(x) = Ke^-α|x|, x ∈ (-∞,∞).

The value of K is

1. 0.5
2. 1
3. 0.5α
4. α

Answer: c

26. Consider the continous random variable with probability density function

f(t) = 1 + t for -1 ≤ t ≤ 0

f(t) = 1 – t for 0 ≤ t ≤ 1

The standard deviation of the random variable is

a. \frac{1}{\sqrt{3}}\\

b. \frac{1}{\sqrt{6}}\\

c. \frac{1}{3}\\

d. \frac{1}{6}]

Answer: b

27. Let f(x) be the continuous probability density function of a random variable X. The probability that a < X ≤ b, is

1. f(b-a)
2. f(b)-f(a)
3. [Image: \int_{a}^{b}f(x)dx]
4. [Image: \int_{a}^{b}xf(x)dx]

Answer: c

28. A point is randomly selected with uniform probability in the X-Y plane within the rectangle with corners at (0, 0), (1,0), (1, 2) and (0, 2). If p is the length of the position vector of the point, the expected value of p² is

1. 2/3
2. 1
3. 4/3
4. 5/3

Answer: d

29. An examination paper has 150 multiple-choice questions of one mark each, with each question having four choices. Each incorrect answer fetches -0.25 mark. Suppose 1000 students choose all their answers randomly with uniform probablity. The sum total of the expected marks obtained by all these students is

1. 2550
2. 0
3. 9375
4. 7525

Answer: c

30. A program consists of two modules executed sequentially. Let f₁(t) and f₂(t) respectively denote the probability density functions of time taken to execute the two modules. The probability density function of the overall time taken to execute the program is given by

1. f₁(t) + f₂(t)
2. [Image: \int_{0}^{t}f_1(x)f_2(x)dx]
3. [Image: \int_{0}^{t}f_1(x)f_2(t-x)dx]
4. max{f₁(t), f₂(t)}

Answer: c

1. 1/2
2. 4/9
3. 5/9
4. 6/9

Answer: a

34. A six-face fair dice is rolled a large number of times. The mean value of the outcomes is

1. 4.5
2. 3.5
3. 2.5
4. 1.5

Answer: b

35. Two coins are tossed simultaneously. The probability (upto two decimal points accuracy) of getting at least one head is

1. 1/4
2. 5/2
3. 3/4
4. 4/3

Answer: c

36. Consider a Poisson distribution for the tossing of a biased coin. The mean for this distribution is μ. The standard deviation for this distribution is given by

1. [Image: \sqrt{\mu}]
2. μ²
3. μ
4. 1/μ

Answer: a

37. The second moment of a Poisson-distributed random variable is 2. The mean of the randor variable is

1. 4
2. 3
3. 2
4. 1

Answer: d

38. The probability that a screw manufactured by a company is defective is 0.1. The company sells screws in packets containing 5 screws and gives a guarantee of replacement if one or more screws in the packet are found to be defective. The probability that a packet would have to be replaced is

1. acceptance of the null hypothesis when it is false and should be rejected
2. rejection of the null hypothesis when it is true and should be accepted
3. rejection of the null hypothesis when it is false and should be rejected
4. acceptance of the null hypothesis when it is true and should be accepted

Answer: a

40. Three cards were drawn from a pack of 52 cards. The probability that they are a king, a queen, and a jack is

1. 16/5525
2. 64/2197
3. 3/13
4. 8/16575

Answer: a

41. X and Y are two random independent events. It is known that P(X) =0.40 and P(X ∪ Y^C) = 0.7. Which one of the following is the value of P(X ∪ Y)?

1. 0.7
2. 0.5
3. 0.4
4. 0.3

Answer: a

42. An urn contains 5 red and 7 green balls. A ball is drawn at random and its colour is noted. The ball is placed back into the urn along with another ball of the same colour. The probability of getting a red ball in the next draw is

1. 65/156
2. 67/156
3. 79/156
4. 89/156

Answer: a

1. 0.001
2. 0.057
3. 0.107
4. 0.3

Answer: b

44. The probability of obtaining at least two “SIX” in throwing a fair dice 4 times is

1. 125/432
2. 13/144
3. 425/432
4. 19/144

Answer: d

45. Two players, A and B, alternately keep rolling a fair dice. The person to get a six first wins the game. Given that player A starts the game, the probability that A wins the game is

1. 5/11
2. 6/11
3. 1/2
4. 7/13

Answer: b

46. If P(X) = 1/4, P(Y) = 1/3, and P(X ∩ Y) = 1/12, the value of P(Y/X) is

1. 29/50
2. 1/3
3. 1/4
4. 4/25

Answer: b

47. The chance of a student passing an exam is 20%. The chance of a student passing the exam and getting above 90% marks in it is 5%, Given that a student passes the examination, the probability that the student gets above 90% marks is

1. 5/18
2. 1/18
3. 2/9
4. 1/4

Answer: d

48. The number of accidents occurring in a plant in a month follows Poisson distribution with mean as 5.2. The probability of occurrence of less than 2 accidents in the plant during a randomly selected month is

1. 0.034
2. 0.029
3. 0.044
4. 0.039

Answer: a

49. A machine produces 0, 1 or 2 defective pieces in a day with associated probability of 1/6, 2/3 and 1/6, respectively. The mean value and the variance of the number of defective pieces produced by the machine in a day, respectively, are

1. 1 and 1/3
2. 1/3 and 1
3. 1 and 4/3
4. 1/3 and 4/3

Answer: a

50. An unbiased coin is tossed an infinite number of times. The probability that the fourth head appears at the tenth toss is

1. 0.067
2. 0.073
3. 0.082
4. 0.091

Answer: c

51. Suppose that a shop has an equal number of LED bulbs of two different types. The probability of an LED bulb lasting more than 100 hours given that it is of Type 1 is 0.7, and given that it is of Type 2 is 0.4. The probability that an LED bulb chosen uniformly at random lasts more than 100 hours is

1. 0.55
2. 0.30
3. 0.60
4. 0.22

Answer: a

52. Consider the following experiment.

Step 1. Flip a fair coin twice.

Step 2. If the outcomes are (TAILS, HEADS) then output Y and stop.

Step 3. If the outcomes are either (HEADS, HEADS) or (HEADS, TAILS), then output N and stop.

Step 4. If the outcomes are (TAILS, TAILS), then go to Step 1.

The probability that the output of the experiment is Y is

(up to two decimal places)

1. 0.66
2. 0.33
3. 0.55
4. 0.76

Answer: b

53. The probability of getting a “head” in a single toss of a biased coin is 0.3. The coin is tossed repeatedly till a “head” is obtained. If the tosses are independent, then the probability of getting “head” for the first time in the fifth toss is

1. 0.072
2. 0.333
3. 0.064
4. 0.234

Answer: a

54. Two coins R and S are tossed. The 4 joint events H_RH_S, T_RT_S, H_RT_S, T_RH_S have probabilities 0.28, 0.18, 0.30, 0.24, respectively, where H represents head and T represents tail. Which one of the following is TRUE?

1. The Coin tosses are independent
2. R is fair, S is not
3. S is fair, A is not
4. The coin tosses are dependent

Answer: d

55. Three vendors were asked to supply a very high precision component. The respective probabilities of their meeting the strict design specification are 0.8, 0.7 and 0.5. Each vendor supplies one component. The probability that out of total three components supplied by the vendors, at least one will meet the design specification is

1. 0.90
2. 0.97
3. 0.87
4. 0.67

Answer: b

56. Suppose A and B are two independent events with probabilities P(A) ≠ 0 and P(B) ≠ 0. Let A’ and B’ be their complements. Which one of the following statements is FALSE?

1. P(A ∩ B) = P(A) P(B)
2. P(A/B) = P(A)
3. P(A ∪ B) = P(A) + P(B)
4. P(A’ ∩ B’) = P(A’) P(B’)

Answer: c

57. The security system at an IT office is composed of 10 computers of which exactly four are working. To check whether the system is functional, the officials inspect four of the computers picked at random (without replacement). The system is deemed functional if at least three of the four computers inspected are working. Let the probability that the system is deemed functional be denoted by p. Then 100p =

1. 12.90
2. 11.90
3. 13.90
4. 10.90

Answer: b

58. Let S be a sample space and two mutually exclusive events A and B be such that A ∪ B = S. If P(·) denotes the probability of the event, the maximum value of P(A) P(B) is

1. 0.75
2. 0.50
3. 0.25
4. 0.20

Answer: c

59. The probability that a given positive integer lying between 1 and 100 (both inclusive) is NOT divisible by 2, 3 or 5 is

60. Four fair six-sided dice are rolled. The probability that the sum of the results being 22 is X/1296. The value of X is

1. 14
2. 13
3. 11
4. 10

Answer: d

63. Parcels from sender S to receiver R pass sequentially through two post-offices. Each post-office has a probability 1/5 of losing an incoming parcel, independently of all other parcels. Given that a parcel is lost, the probability that it was lost by the second post-office is

64. In a housing society, half of the families have a single child per family, while the remaining half have two children per family. The probability that a child picked at random, has a sibling is

65. Consider a dice with the property that the probability of a face with n dots showing up is proportional to n. The probability of the face with three dots showing up is

66. A fair coin is tossed n times. The probability that the difference between the number of heads and tails is (n-3) is

1. 2^-n
2. 0
3. 2^-n+3
4. ⁿC_n-32^-n

Answer: b

67. A fair coin is tossed till a head appears for the first time. The probability that the number of required tosses is odd, is

1. 1/3
2. 1/2
3. 2/3
4. 3/4

Answer: c

68. A box contains 4 white balls and 3 red balls. In succession, two balls are randomly selected and removed from the box. Give that the first removed ball is white, the probability that the second removed ball is red is

1. 1/3
2. 3/7
3. 1/2
4. 4/7

Answer: c

69. A loaded dice has following probability distribution of occurrences

If three identical dice as the above are thrown, the probability of occurrence of values 1, 5 and 6 on the three dice is

1. same as that of occurrence of 3, 4, 5
2. same as that of occurrence of 1, 2, 5
3. 1/128
4. 5/8

Answer: c

70. Two fair dice are rolled and the sum r of the numbers turned up is considered

1. Pr(r > 6) = (1/6)
2. Pr(r/3 is an integer) = (5/6)
3. Pr(r = 8 | r/4 is an integer) = (5/9)
4. Pr(r= 6 | r/5 is an integer) = (1/18)

Answer: c

71. If P and Q are two random events, then the following is TRUE

1. Independence of P and Q implies that probability (P ∩ Q) = 0
2. Probability (P ∪ Q) ≥ Probability (P) + Probability (Q)
3. If P and Q are mutually exclusive, then they must be independent
4. Probability (P ∩ Q) ≤ Probability (P)

Answer: d

72. A fair coin is tossed three times in succession. If the first toss produces a head, then the probability of getting exactly two heads in three tosses is

1. 1/8
2. 1/2
3. 3/8
4. 3/4

Answer: b

73. A group consists of equal number of men and women. Of this group 20% of the men and 50% of the women are unemployed. If a person is selected at random from this group, the probability of the selected person being employed is

74. A box contains 25 parts of which 10 are defective. Two parts are being drawn simultaneously in a random manner from the box. The probability of both the parts being good is

1. 5/9
2. 25/29
3. 45/125
4. 7/20

Answer: d

75. A nationalized bank has found that the daily balance available in its savings accounts follows a normal distribution with a mean of Rs. 500 and a standard deviation of Rs. 50. The percentage of savings account holders, who maintain an average daily balance more than Rs. 500 is

1. 100
2. 50
3. 75
4. 25

Answer: b

76. A box contains 4 red balls and 6 black balls. Three balls are selected randomly from the box one after another, without replacement. The probability that the selected set contains one red ball and two black balls is

1. 1/20
2. 1/12
3. 3/10
4. 1/2

Answer: d

77. An unbiased coin is tossed five times. The outcome of each toss is either a head or a tail. The probability of getting at least one head is

1. 31/32
2. 16/32
3. 13/32
4. 1/32

Answer: a

78. A box contains 2 washers, 3 nuts and 4 bolts. Items are drawn from the box at random one at a time without replacement. The probability of drawing 2 washers first followed by 3 nuts and subsequently the 4 bolts is

1. 1/630
2. 2/315
3. 1/2520
4. 1/1260

Answer: d

79. If three coins are tossed simultaneously, the probability of getting at least one head is

1. 3/8
2. 1/8
3. 7/8
4. 1/2

Answer: c

80. A coin is tossed 4 times. What is the probability of getting heads exactly 3 times?

1. 1/4
2. 3/8
3. 1/2
4. 3/4

Answer: a

81. A box contains 20 defective items and 80 non-defective items. If two items are selected at random without replacement, what will be the probability that both items are defective?

1. 19/495
2. 20/99
3. 1/25
4. 1/5

Answer: a

82. A lot has 10% defective items. Ten items are chosen randomly from this lot. The probability that exactly 2 of the chosen items are defective is

1. 0.0036
2. 0.1937
3. 0.3874
4. 0.2234

Answer: b

83. A single die is thrown twice. What is the probability that the sum is neither 8 nor 9?

1. 1/9
2. 5/36
3. 1/4
4. 3/4

Answer: d

84. From a pack of regular playing cards, two cards are drawn at random. What is the probability that both cards will be Kings, if first card in NOT replaced?

1. 1/221
2. 1/169
3. 1/52
4. 1/26

Answer: a

85. A box contains 5 black and 5 red balls. Two balls are randomly picked one after another from the box, without replacement. The probability for both balls being red is

1. 1/90
2. 1/2
3. 19/90
4. 2/9

Answer: d

86. A fair (unbiased) coin was tossed four times in succession and resulted in the following outcomes: (i) Head, (ii) Head, (iii) Head, (iv) Head. The probability of obtaining a ‘Tail’ when the coin is tossed again is

1. 0
2. 1/2
3. 4/5
4. 1/5

Answer: b

87. The annual precipitation data of a city is normally distributed with mean and standard deviation as 1000 mm and 200 mm, respectively. The probability that the annual precipitation will be more than 1200 mm is

1. <50%
2. 50%
3. 75%
4. 100%

Answer: a

88. In an experiment, positive and negative values are equally likely to occur. The probability of obtaining at most one negative value in five trials is

1. 6/32
2. 3/32
3. 2/32
4. 1/32

Answer: a

89. There are two containers, with one containing 4 red and 3 green balls and the other containing 3 blue and 4 green balls. One ball is drawn at random from each container. The probability that one of the balls is red and the other is blue will be

1. 1/7
2. 9/49
3. 12/49
4. 3/7

Answer: c

90. A person on a trip has a choice between private car and public transport. The probability of using a private car is 0.45. While using the public transport, further choices available are bus and metro, out of which the probability of commuting by a bus is 0.55. In such a situation, the probability (rounded up to two decimals) of using a car, bus and metro, respectively would be

1. 0.45, 0.30 and 0.25
2. 0.45, 0.25 and 0.30
3. 0.45, 0.55 and 0.00
4. 0.45, 0.35 and 0.20

Answer: a

93. A box contains 10 screws, 3 of which are defective. Two screws are drawn at random with replacement. The probability that none of the two screws is defective will be

1. 100%
2. 50%
3. 49%
4. None of these

Answer: c

94. Suppose p is the number of cars per minute passing through a certain road junction between 5 PM, and p has Poisson distribution with mean 3. What is the probability of observing fewer than 3 cars during any given minute in this interval?

1. 8/(2e³)
2. 9/(2e³)
3. 26/(2e³)
4. 17/(2e³)

Answer: d

1. 5/12
2. 10/21
3. 1/6
4. 2/3

Answer: a

96. Consider the finite sequence of random values X= [x₁, x₂, ….., x_n]. Let μ_x be the mean and σ_x, be the standard deviation of X. Let another finite sequence Y of equal length be derived from this as y_i = a*x_i, + b, where a and b are positive constant. Let μ_y be the mean and σ_y be the standard deviation of this sequence. Which one of the following statements INCORRECT?

1. Index position of mode of X in X is the same as the index position of mode of Y in Y.
2. Index position of median of X in X is the same as the index position of median of Y in Y
3. μ_y = aμ_x + b
4. σ_y = aσ_x + b

Answer: d

97. A deck of 5 cards (each carrying a distinct number from 1 to 5) is shuffled thoroughly. Two cards are then removed one at a time from the deck. What is the probability that the two cards are selected with the number on the first card being one higher than the number on the second card?

1. 1/5
2. 4/25
3. 1.4
4. 2/5

Answer: a

98. Consider a company that assembles computers. The probability of a faulty assembly of any computer is p. The company, therefore, subjects each computer to a testing process. This testing process gives the correct result for any computer with a probability of q. What is the probability of a computer being declared faulty?

1. pq + (1- p) (1- q)
2. (1-q)p
3. (1-p)q
4. pq

Answer: a

99. If two fair coins are flipped and at least one of the outcomes is known to be a head, what is the probability that both outcomes are heads?

1. 1/4
2. 1/3
3. 2/3
4. 1/2

Answer: b

100. An unbalanced dice (with 6 faces, numbered from 1 to 6) is thrown. The probability that the face value is odd is 90% of the probability that the face value is even. The probability of getting any even-numbered face is the same. If the probability that the face is even given that it is greater than 3 is 0.75, which one of the following options is closest to the probability that the face value exceeds 3?

1. 0.453
2. 0.468
3. 0.485
4. 0.492

Answer: b

1. 0.24
2. 0.36
3. 0.4
4. 0.6

Answer: c

102. Suppose we uniformly and randomly select a permutation from the 20! permutations of 1, 2, 3, ……, 20. What is the probability that 2 appears at an earlier position that any other even number in the selected permutation?

1. 1/2
2. 1/10
3. 9!/20!
4. None of these

Answer: d

103. Two n bit binary strings, S1 and S2 are chosen randomly with uniform probability. The probability that the Hamming distance between these strings (the number of bit positions where the two strings differ) is equal to d is

1. ⁿC_d/2ⁿ
2. ⁿC_d/2^d
3. d/2ⁿ
4. 1/2^d

Answer: a

104. If a fair coin is tossed four times. What is the probability that two heads and two tails will result?

1. 3/8
2. 1/2
3. 3/4
4. 5/8

Answer: a

1. 1/4, 1/2
2. 1/2, 1/4
3. 1/2, 1
4. 1, 1/2

Answer: d

106. A fair dice is tossed two times. The probability that the second toss results in a value that is higher than the first toss is

1. 2/6
2. 2/36
3. 1/2
4. 5/12

Answer: d

107. A fair coin is tossed independently four times. The probability of the event ”the number of times heads show up is more than the number of times tails show up” is

1. 1/16
2. 1/8
3. 1/4
4. 5/16

Answer: d

108. A fair coin is tossed 10 times. What is the probability that only the first two tosses will yield heads?

1. (1/2)²
2. ¹⁰C₂(1/2)²
3. (1/2)¹⁰
4. ¹⁰C₂(1/2)¹⁰

Answer: c

109. An examination consists of two papers, Paper 1 and Paper 2. The probability of failing in Paper 1 is 0.3 and that in Paper 2 is 0.2. Given that a student has failed in Paper 2, the probability of failing in Paper 1 is 0.6. The probability of a student failing in both the papers is

1. 0.18
2. 0.5
3. 0.12
4. 0.06

Answer: c

110. Three companies X, Yand Z supply computers to a university. The percentage of computers supplied by them and the probability of those being defective are tabulated below:

Given that a computer is defective, the probability that it was supplied by Y is

1. 0.2
2. 0.1
3. 0.3
4. 0.4

Answer: d

111. A fair dice is rolled twice. The probability that an odd number will follow an even number is

1. 1/2
2. 1/6
3. 1/3
4. 1/4

Answer: d

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