AGNI 4.

0
🔥Jab tak todega nahi chodega nahi🔥
DPP – 9 (Probability)

1. Three unbiased coins are tossed. The probability of 9. For three events 𝐴, 𝐵 and 𝐶, 𝑃( Exactly one of 𝐴 or 𝐵
getting at least 2 tails is occurs ) = P( Exactly one of B or C occurs ) = P(
3 1 1 1 1
a. 4
b. 4
c. 2
d. 3
Exactly one of C or A occurs ) = 4 and P(All the three
1
2. A single letter is selected from the word TRICKS. events occur simultaneously) = 16. Then the probability
The probability that it is either T or R is that at least one of the events occurs, is
1 1 1 1 7 7 3 7
a. b. c. d. a. b. c. d.
36 4 2 3 16 64 16 32

3. Two dice are thrown simultaneously. What is the 10. If 𝑥 is one of the first fifty numbers chosen at
probability of getting two numbers whose product is 3
random, then the probability that 𝑥 + 𝑥 is greater than
even?
20 is
3 1 1 2
a. b. c. d. 11 21 31 41
4 4 2 3 a. b. c. d.
50 50 50 50

4. The probability that a non leap year selected at

11. Two distinct numbers 𝑥 and 𝑦 are chosen from
random will have 53 Sundays, is
1,2,3,4,5. The probability that the arithmetic mean of 𝑥
a. 0 b. 1/7 c. 2/7 d. 3/7
and 𝑦 is an integer is
5. Two fair dice are rolled. The probability of getting a 1 3 2
a. 0 b. c. d.
5 5 5
composite number as the sum of face values is equal
to 12. In a class of 15 students, 5 of them are boys and 10
7 5 1 3 students are girls. A team of 11 members has to be
a. b. c. d.
12 12 12 4
formed at random. The probability that the team has at
6. Three dice are rolled once. The chance of getting a least 4 boys, is
total score of 5 is 57 54 54 51
a. 91
b. 95
c. 91
d. 91
5 1 1 1
a. 216
b. 6
c. 36
d. 72
13. Two cards are drawn at random from a pack of 52
7. Probability of product of a perfect square when two cards. The probability of these two being "Aces" is
dice are thrown together is ...... 1 1 1 1
a. 26
b. 221
c. 2
d. 13
1 2 2 4
a. 9
b. 13
c. 9
d. 9
14. A bag contains 3 red, 4 white and 5 blue balls. If
8. Twelve tickets are numbered from 1 to 12 . One two balls are drawn at random, then the probability that
ticket is drawn at random, then the probability of the they are of different colours is
number to be divisible by 2 or 3 is 47 23 47 47
a. b. c. d.
66 33 132 33
a. 2/3 b. 7/12 c. 5/6 d. ¾
15. There are 5 positive numbers and 6 negative
numbers. Three numbers are chosen at random and
Prepared by Anuj Seth Contact 9310913371
 AGNI 4.0
🔥Jab tak todega nahi chodega nahi🔥
DPP – 9 (Probability)

multiplied. The probability that the product being a numbers have x20 as the middle number is
20 C 30 C 30 C 19 C 19 C 31
negative number is 2× 2 2× 2 2 × C2
a. 50 C b. 50 C c. 50 C d. None of these
11 17 16 16 5 5 5
a. 34
b. 33
c. 35
d. 33
22. Let A and B be two events. Then 1 + P(A ∩ B) −
16. An urn contains 9 balls, 2 of which are white, 3 P(B) − P(A) is equal to
blue and 4 black are drawn at random from the urn. ̅∪B
a. P(A ̅) b. P(A ∩ B
̅) c. P(A
̅ ∩ B) d. P(A
̅∩B
̅)
The chance that 2 balls will be of the same colour and
23. If the occurrence of an event 𝐴 implies the
the third of a different colour is
45 55 35 25 occurrence of an event B, then P(AC ∩ BC ) is equal to
a. 84
b. 84
c. 84
d. 84
a. 𝑃(𝐵𝐶 ) b. P(AC )P(BC ) c. 𝑃 (𝐴𝐶 ) d. 1 − P(A ∩ B)
17. 6 boys & 6 girls sit in a row at random. The
24. Consider an experiment E in which a box contains
probability that all the girls sit together is
1 12 1
10 identical tickets numbered 1 to 10 and 2 tickets are
a. b. c. d. None of these
432 431 132 drawn at random from the box. What is the probability
18. A bag contains 3 white, 4 black, 2 red balls. If 2 that both the tickets have even numbers on them?
4 1 2 1
balls are drawn at random, then the probability that a. b. c. d.
9 3 9 9
both the balls are white, is
1 1 1 1
25. If 𝐴, 𝐵 and 𝐶 are mutually exclusive and exhaustive
a. 18
b. 36
c. 12
d. 24 3
events of a random experiment such that P(B) = 2 P(A)
19. Two persons 𝐴 and 𝐵 are throwing an unbiased six 1
and P(C) = 2 (P(B), then P(A ∪ C) =
faced die alternatively, with the condition that the
3 6 7 10
a. b. c. d.
person who throws 3 first wins the game. If A starts the 13 13 13 13

game, the probabilities of A and B to win the same are 26. There are 7 horses in a race. Mr. X selected 2
respectively horses at random and bet on them. The probability that
6 5 5 6 8 3 3 8
a. 11 11
, b. 11 11
, c. 11 11
, d. 11 11
, Mr. X selected the winning horse, is
1 4 3 2
a. b. c. d.
20. The letters of the word "Question" are arranged in a 7 7 7 7

row at random. The probability that there are exactly 3

27. The probability that A speaks the truth is 5 and
two letters between 𝑄 and 𝑆 is
3
1 5 1 5 probability that B speaks the truth is . Find the
4
a. b. c. d.
14 7 7 28
probability that they contradict each other when asked
21. x1 , x2 , x3 , … x50 are fifty real numbers such that xr < to speak a fact.
xr+1 for 𝑟 = 1,2,3, … .49. Five numbers out of these are a.
3
b.
4
c.
9
d.
7
20 5 20 20
picked up at random. The probability that the five

Prepared by Anuj Seth Contact 9310913371

 AGNI 4.0
🔥Jab tak todega nahi chodega nahi🔥
DPP – 9 (Probability)

1 1
28. For any two events A, B if P(A) = 3 , P(B) = 2, P(A ∩ least one of them is
235 21 3 256
1 a. b. c. d.
B) = 6, then the probability that exactly one of them 256 256 256 256

occurs is 34. A bag contains 50 tickets numbered 1,2,3, … . ,50 of

1 1 2 5
a. b. c. d. which five are drawn at random and arranged in
3 2 3 6
ascending order of magnitude (x1 < x2 < x3 < x4 < x5 )
29. A four-digit number is formed by the digits 1,2,3,4
then the probability that x3 = 30 is
with no repetition. The probability that the number is 29 C 20 C 30 C 20 C 5C 50 C 50 C 20 C
2× 2 1× 1 1× 2 2× 1
a. 50 C b. 50 C c. 50 C d. 50 C
odd, is 5 5 5 5

1 1 1
a. zero b. c. d. None of these 35. If S is the sample space and P(A) = 3 P(B) and S =
3 4
A ∪ B where A and B are two mutually exclusive events,
30. A poker hand consists of 5 cards drawn at random
then P(A) = ?
from a well-shuffled pack of 52 cards. Then the
a. ¼ b. ½ c. ¾ d. 3/8
probability that the pocket hand consists of a pair and a
36. Five persons entered the lift cabin on the ground
triple of equal face values (for example, 2 sevens and 3
floor of an eight floor house. Suppose that each of them
kings or 2 aces and 3 queens, etc.) is
6 23 1797 1 independently and with equal probability can leave the
a. 4165
b. 4165
c. 4165
d. 4165
cabin at any floor beginning with the first, then the
31. For the two events A and B, let P(A) = 0.7 and probability of all 5 persons leaving at different floors is
7P 75 5P
P(B) = 0.6. The necessarily false statement(s) is/are a. 5
b. c.
6
d. 5
75 7𝑃
5
6P
5 55
a. P(A ∩ B) = 0.35 b. P(A ∩ B) = 0.45
c. P(A ∩ B) = 0.65 d. P(A ∩ B) = 0.28 37. Five dice are tossed. What is the probability that the
five numbers shown will be different?
32. For any two events 𝐴 and 𝐵, which of the following 5 5 5 5
a. 54
b. 18
c. 27
d. 81
result does not hold true in general:
a. P(A ∪ B) = P(A) + P(B) − P(A ∩ B) 38. An objective type test paper has 5 questions. Out of
b. 𝑃(𝐴) = 𝑃(𝐴 ∩ 𝐵) + 𝑃(𝐴 ∩ 𝐵‾) these 5 questions, 3 questions have four options each
̅ ∩ B)
c. P(B) = P(A ∩ B) + P(A (A, B, C, D) with one option being the correct answer.
d. 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) The other 2 questions have two options each, namely
true and false. A candidate randomly ticks the options.
33. Four persons independently solve a certain
1 3 1 1 Then the probability that he/she will tick the correct
problem correctly with probabilities , , , . Then the
2 4 4 8
option in atleast four questions is
probability that the problem is solved correctly by at 5 3 3 3
a. 32
b. 128
c. 256
d. 64

Prepared by Anuj Seth Contact 9310913371

 AGNI 4.0
🔥Jab tak todega nahi chodega nahi🔥
DPP – 9 (Probability)

39. Each of a and b can take values of 1 or 2 with equal at least one pair, is
probability. The probability that the equation ax 2 + bx + a. 41 b. 120
c.
21
d. None of these
161 161 161
1 = 0 has real roots, is equal to
1 1 1 1
47. 4 boys and 2 girls occupy seats in a row at random.
a. b. c. d.
2 4 8 16 Then the probability that the two girls occupy seats side
40. The probability that atleast one of the events 𝐴 and by side is
1 1 1 1
𝐵 occurs is 0.5 . If 𝐴 and 𝐵 occur simultaneously with a. b. c. d.
2 4 3 6
probability 0.2 , then P(AC ) + P(BC ) is equal to
48. Three cards are drawn successively without
a. 1.0 b. 1.1 c. 0.7 d. 1.3
replacement from a pack of 52 well shuffled cards. The
41. Let 𝐴 and 𝐵 two mutually exclusive events such that probability that first two cards are queens and the third
P(A ∩ BC ) = 0.25 and P(A C ∩ B) = 0.5. Then P((A ∪ car is a king is
4 4 4 4 2 1
B)C ) is equal to a. × × b. × ×
52 51 50 52 51 50
a. 0.25 b. 0.50 c. 0.75 d. 0.40 4 3
c. 52 × 51 × 50
3
d.
4 3
× 51 × 50
4
52
42. The probability that atleast one of the events 𝐴 and
49. An urn contains nine balls of which three are red,
𝐵 occurs is 0.6 . If 𝐴 and 𝐵 occur simultaneously with
four are blue and two are green. Three balls are drawn
probability 0.2 , then 𝑃(𝐴‾) + 𝑃(𝐵‾) is
at random without replacement from the urn. The
a. 0.4 b. 0.8 c. 1.2 d. 1.4
probability that the three balls have different colours is
43. An urn contains 8 red and 5 white balls. Three balls a. 1/3 b. 2/7 c. 1/21 d. 2/23
are drawn at random. Then the probability that balls of
50. Two events 𝐴 and 𝐵 have probabilities 0.3 and 0.4
both colours are drawn is
respectively. The probability that both A and B occur
40 70 3 10
a. b. c. d.
143 143 13 13 simultanesously is 0.1 . The probability that neither A

44. Let A and B abe any two events, then P(A ∩ B) is nor B occur is

equal to a. 0.2 b. 0.3 c. 0.4 d. 0.1

a. 𝑃(𝐴 ∪ 𝐵) − 𝑃(𝐴𝐶 ) b. P(A) + P(BC ) 51. If 𝐴 and 𝐵 are mutually exclusive events and if
𝐶) 1 13
c. 𝑃(𝐵) + 𝑃(𝐴 d. None of the above 𝑃(𝐵) = 3, P(A ∪ B) = 21, then P(A) is equal to

45. If 𝐴 and 𝐵 are mutually exclusively events such that a. 1/7 b. 4/7 c. 2/7 d. 5/7
P(A) = 0.25, P(B) = 0.4, then P(AC ∩ BC ) is equal to 52. 7 persons to be seated in a row. Probability that 2
a. 0.45 b. 0.55 c. 0.9 d. 0.35 particular persons to sit next to each other is
46. A bag contains 12 pairs of socks, 4 socks are a. 3/7 b. 2/7 c. 4/7 d. 5/7
picked up at random. Then, the probability that there is
Prepared by Anuj Seth Contact 9310913371
 AGNI 4.0
🔥Jab tak todega nahi chodega nahi🔥
DPP – 9 (Probability)

53. In a non-leap year, the probability of having 53 60. In a college 25% boys and 10% girls offer
Friday or Saturday is Mathematics. There are 60% girls in the college. If a
a. 3/7 b. 4/7 c. 2/7 d. 1/7 Mathematics student is chosen at random, then the
1+3𝑝 1−𝑝 1−2𝑝 probability that the student is a girl, will be
54. If 3
, 4
and 2
are mutually exclusive events.
1 3 5 5
a. 6
b. 8
c. 8
d. 6
Then, range of p is
1 1 1 1 1 2 1 2
a. 3 ≤ p ≤ 2 b. ≤ p ≤ 2 c. ≤ p ≤ 3 d. ≤p≤5 61. In an assembly of 4 persons the probability that at
4 3 3
least 2 of them have the same birthday, is
3
55. If A and B are two events such that P(A) = and a. 0.293 b. 0.24 c. 0.0001 d. 0.016
4
5
P(B) = 8, then 62. 𝐴 and 𝐵 are two independent events such that
a. P(A ∪ B) ≥ 3/4 b. P(A′ ∩ B) ≤ 1/4 𝑃 (𝐴 ∪ 𝐵′ ) = 0.8 and P(A) = 0.3. Then P(B) is
3 5
c. 8 ≤ P(A ∩ B) ≤ 8 d. All of the above a. 2/7 b. 2/3 c. 3/8 d. 1/8

56. Two dice are tossed once. The probability of getting 63. Three numbers are chosen at random from 1 to 20 .
an even number at the first die or a total of 8 is The probability that they are consecutive is
1 1 3 5
a.
1
b.
3
c.
11
d. None of these a. 190
b. 120
c. 190
d. 190
36 36 36

57. The probability that at least one of A and B occurs 64. The probability that A can solve a problem is 2/3

is 0.6 . If 𝐴 and 𝐵 occur simultaneously with probability and 𝐵 can solve it is 3/4. If both attempt the problem,
0.3 , then P(A′ ) + P(B′ ) is what is the probability that the problem gets solved?
11 7 5 1
a. 0.9 b. 0.15 c. 1.1 d. 1.2 a. 12
b. 12
c. 12
d. 2

58. A complete cycle of a traffic light takes 60 seconds. 65. If 𝐴 and 𝐵 are two events such that 𝑃(𝐴) =
During each cycle the light is green for 25 seconds, 0.54, 𝑃(𝐵) = 0.69 and P(A ∩ B) = 0.35, then P(A ∩ B′ ) is
yellow for 5 seconds and red for 30 seconds. At a a. 0.88 b. 0.12 c. 0.19 d. 0.34
randomly chosen time, the probability that the light will
66. Two events 𝐴 and 𝐵 have probabilities 0.25 and
not be green is
0.50 respectively. The probability that both 𝐴 and 𝐵
1 1 4 7
a. b. c. d.
3 4 12 12 occur is 0.14 . Then the probability that neither nor 𝐵

59. A five digit number is formed by the digits 1,2,3,4,5,6 occur is

& 8. The probability that the no. has even digits at both a. 0.39 b. 0.25 c. 0.11 d. None of these

ends is 67. A die is thrown. Let A be the event that the number
a. 2/7 b. 3/7 c. 4/7 d. 1/7 obtained is greater than 3 . Let 𝐵 be the event that the

Prepared by Anuj Seth Contact 9310913371

 AGNI 4.0
🔥Jab tak todega nahi chodega nahi🔥
DPP – 9 (Probability)

number obtained is less than 5 . Then P(A ∪ B) is 75. When three dice are thrown the probability of 4 or 5
a. 2/5 b. 3/5 c. 0 d. 1 on each of the dice simultaneously is
a. 1/72 b. 1/108 c. 1/24 d. None of these
68. Let 𝐸1 , 𝐸2 be two mutually exclusively events of an
76. The probability of forming a three digit number with
experiment with P (not E2 ) = 0.6 = P(E1 ∪ E2 ). Then
the same digits when three digit numbers are formed
P(E1 ) =
out of the digits 0,2,4,6,8
a. 0.1 b. 0.3 c. 0.4 d. 0.2
a. 1/16 b. 1/12 c. 1/645 d. 1/25
69. A die has four blank faces and two faces marked 3.
1
̅ ) = 2, find P(A
77. P(A ∪ B) = , P(A ̅ ∩ B)
The chance of getting a total of 12 in 5 throws is 2 3

1 4 2 1 2 4 a. 1/3 b. ¼ c. 1/5 d. 1/6

a. 5 C4 (3) (3) b. 5
C4 (3) (3)

1 5 1 4 5 78. If 𝐴 and 𝐵 are mutually exclusive events with

5 5
c. C4 (6) d. C4 (6) (6) 1
𝑃(𝐴) = 2 𝑃(𝐵) and A ∪ B = S, the sample space then
70. If 𝐴 and 𝐵 are any two events, then 𝑃 (𝐴 ∩ 𝐵′ ) P(A) =
a. P(A) + P(B′ ) b. P(A) + P(A ∩ B)
a. 2/3 b. 1/3 c. ¼ d. ¾
c. P(B) − P(A ∩ B) d. P(A) − P(A ∩ B)
79. A coin is tossed three times. The probability of
71. Let A = {1,3,5,7,9}, B = {2,4,6,8}, if a cartesian getting a head once and a tail twice is
product A × B choosen at random, the probability of a + a. 1/8 b. ¼ c. 3/8 d. ½
b = 9 is
80. The probability of choosing a number divisible by 6
a. 3/2 b. ¾ c. 1 d. 1/5
or 8 from among 1 to 90 is
3
72. If 𝐴 and 𝐵 are two events such that 𝑃(𝐴 ∪ 𝐵) = 4, a. 1/6 b. 1/90 c. 1/30 d. 23/90
1 2
̅ ) = , then P(A
P(A ∩ B) = 4 , P(A ̅ ∩ B) is equal to 81. Out of 15 persons, 10 can speak Hindi and 8 can
3
5 3 5 1 speak English. If two persons are chosen at rando,
a. 12
b. 8
c. 8
d. 4
then the probability that one person speaks Hindi only
73. If four dice are thrown together. Probability that the and the other speaks both Hindi and English is
sum of the numbers appearing on them is 13 , is a. 3/5 b. 7/12 c. 1/5 d. 2/5
35 5 11 11
a. 324
b. 216
c. 216
d. 432 82. An urn contains 3 red and 5 blue balls. The
74. Three different integers are chosen at random from probability that 2 nd ball drawn is blue without
the first 20 integers. The probability that their product is replacement is
even is a. 5/8 b. 8/5 c. 3/8 d. None of these
2 3 17 4
a. b. c. d.
19 19 19 29

Prepared by Anuj Seth Contact 9310913371

 AGNI 4.0
🔥Jab tak todega nahi chodega nahi🔥
DPP – 9 (Probability)

83. A problem is given to three persons and their is equal to

1 1 1 11 5 8 1
chances of solving it are 3 , 5 , 6 respectively. The a. b. c. d.
20 17 17 4

probability that none will solve it is 7 17

90. If P(A ∩ B) = 10 and P(B) = 20, where P stands for
1 1 1 2 4 5 2 4 5 1 1 1
a. 3 × 5 × 6 b. 3 × 5 × 6 c. 1 − 3 × 5 × 6 d. 3 + 3 + 6
probability then P(A ∣ B) is equal to
84. One card is drawn from a pack of 52 cards. The 7 17 14 1
a. 8
b. 20
c. 17
d. 8
probability that it is the card of a king or spade is
91. A fair die is rolled. Consider the events A =
a. 1/26 b. 3/26 c. 4/13 d. 3/13
{1,3,5}, B = {2,3} and C = {2,3,4,5}. Then the conditional
85. A box contains 9 tickets numbered 1 to 9 inclusive.
probability P((A ∪ B) ∣ C) is
If 3 tickets are drawn from the box one at a time, the 1 5 1 3
a. b. c. d.
probability that they are alternatively either {odd, even, 4 4 2 4

odd or {even, odd, even} is 92. Given that 𝑃(𝐴) = 0.1, 𝑃(𝐵 ∣ 𝐴) = 0.6 and 𝑃(𝐵 ∣
5 4 5 5
a. b. c. d. 𝐴𝐶 ) = 0.3 what is 𝑃(𝐴 ∣ 𝐵) ?
17 17 16 18
2 4 7 9
a. b. c. d.
86. Let S be the set of all 2 × 2 symmetric matrices 11 11 11 11

whose entries are either zero or one. A matrix X is 93. A card is picked at random from a pack of cards.

chosen from S. The probability that the determinant of X Given that the picked card is a Queen, what is the
is not zero probability that it is a spade?
1 4 1 1
a. 1/3 b. ½ c. ¾ d. ¼ a. b. c. d.
3 13 4 2

87. A determinant of second order is made with the 94. A six-faced unbiased die is thrown twice and the
elements 0,1 . What is the probability that the sum of the numbers appearing on the upper face is
determinant is nonnegative? observed to be 7 . The probability that the number 3
7 11 3 15
a. b. c. d. has appeared at least once is
12 12 16 16
a. ½ b. 1/3 c. ¼ d. 1/5
1 1 1 AC
88. If P(A) = 4 , P(B) = 5 and P(AB) = 8, then P (BC ) =
95. 𝐴 and 𝐵 are two events such that 𝑃(𝐴) ≠ 0, 𝑃(𝐵 ∣ 𝐴)
21 25 27 29
a. b. c. d. is i. A is a subset of B ii. A ∩ B = 𝜙 are respectively
32 32 32 32

2 a. 1,1 b. 0 and 1 c. 0,0 d. 1,0

89. If 𝐴 and 𝐵 are any two events such that 𝑃(𝐴) = 5
3 96. If 𝐴 and 𝐵 are any two events associated with a
and P(A ∩ B) = , then the conditional probability,
20 3 5
random experiment such that P(A) = 8 , P(B) = 8 and
(A ∣ (A ∪ B ′ ′) ′
), where A denotes the complement of A,
3
P(A or B) = 4, then P(A ∣ B) is

a. ¼ b. 2/3 c. ¾ d. 2/5

Prepared by Anuj Seth Contact 9310913371

 AGNI 4.0
🔥Jab tak todega nahi chodega nahi🔥
DPP – 9 (Probability)

97. Three numbers are chosen at random without 1 5 1

103. If P(A) = , P(B) = and P(B ∣ A) = , then
12 12 15
replacement from {1,2,3, … . ,8}. The probability that their
P(A ∪ B) is equal to
minimum is 3 , given that their maximum is 6 , is 89 90 91 92
a. 180
b. 180
c. 180
d. 180
a. ¼ b. 2/5 c. 3/8 d. 1/5
104. Let 𝑋 and 𝑌 be two events such that 𝑃(𝑋) =
98. For the married couple living in Jammu, the
1 1 2
probability that a husband will vote in an election is 0.5 3
, 𝑃(𝑋 ∣ 𝑌) = 2 and P(Y ∣ X) = 5. Then
1 1
and the probability that his wife will vote is 0.4 . The a. P(X ∩ Y) = 5 b. P(X ′ ∣ Y) = 2
probability that the husband votes, given that his wife 4 2
c. 𝑃(𝑌) = 15 d. 𝑃(𝑋 ∪ 𝑌) = 5
also votes is 0.7 . Then the probability that husband
and wife both will vote is 105. Three persons, 𝑃, 𝑄 and 𝑅 independently try to hit
a. 0.28 b. 0.20 c. 0.35 d. 0.15 a target. If the probabilities of their hitting the target are
3 1 5
99. Let A and B be two events with P(AC ) = 0.3, P(B) = , and respectively, then the probability that the
4 2 8

0.4 and P(A ∩ BC ) = 0.5. Then P(B ∣ A ∪ BC ) is equal to target is hit by P or Q but not by R is
39 21 15 9
a. ¼ b. 1/3 c. ½ d. 2/3 a. 64
b. 64
c. 64
d. 64

100. If C and D are two events such that C ⊂ D and 106. Let E and F be two independent events. The
P(D) ≠ 0, then the correct statement among the probability that both E and F happen is 12 and the
1

following is 1
probability that neither 𝐸 nor 𝐹 happens is 2, then a
𝑃(𝐷)
a. P(C ∣ D) < P(C) b. 𝑃(𝐶 ∣ 𝐷) = 𝑃(𝐶)
𝑃(𝐸)
value of 𝑃(𝐹) is
c. 𝑃(𝐶 ∣ 𝐷) = 𝑃(𝐶) d. 𝑃(𝐶 ∣ 𝐷) ≥ 𝑃(𝐶)
1 5 3 4
a. b. c. d.
101. A dice is rolled twice and the sum of the numbers 3 12 2 3

appearing on them is observed to be 7 . What is the 107. 𝑃 speaks truth in 70% cases and 𝑄 speaks in 80%
conditional probability that the number 2 has appeared of the cases. In what percentage of cases are they
at least once? likely to contradict each other in stating the same fact?
a. ½ b. 1/3 c. 2/3 d. 2/5 a. 25% b. 38% c. 42% d. 48%

102. One Indian and four American men and their 108. Two events 𝐴 and 𝐵 will be independent if
wives are to be seated randomly around a circular a. P(A′ ∩ B′ ) = (1 − P(A))(1 − P(B))
table. Then the conditional probability that the Indian b. 𝐴 and 𝐵 are mutually exclusive
man is seated adjacent to his wife given that each c. 𝑃(𝐴) + 𝑃(𝐵) = 1 d. 𝑃(𝐴) = 𝑃(𝐵)
American man is seated adjacent to his wife is
109. If 𝐴 and 𝐵 are independent events associated to
a. ½ b. 1/3 c. 2/5 d. 1/5
some experiment 𝐸 such that 𝑃 (𝐴𝐶 ∩ 𝐵) = 2/15 and
Prepared by Anuj Seth Contact 9310913371
 AGNI 4.0
🔥Jab tak todega nahi chodega nahi🔥
DPP – 9 (Probability)

𝑃 (𝐴 ∩ 𝐵𝐶 ) = 1/6, then P(B) is equal to 115. If the events 𝐴 and 𝐵 are independent and if
2
a.
1 1
, b.
1 4
, c.
4 1
, d.
4 5
, ̅) = 2, then P(A ∩ B) is equal to
𝑃(𝐴‾) = , P(B
6 5 6 5 5 5 5 6 3 7
4 3 5 2
110. If A and B are independent events such that a. 21
b. 21
c. 21
d. 21
2
̅) = 0.8, then P(A) =
P(B) = 7, P(A ∪ B 116. Let 𝑋 and 𝑌 be two events such that 𝑃(𝑋 ∣ 𝑌) =
1 1 1
a. 0.4 b. 0.3 c. 0.2 d. 0.1 , 𝑃(𝑌 ∣ 𝑋) = 3 and P(X ∩ Y) = 6. Which of the following
2

111. If 𝐴 and 𝐵 are two independent events such that is (are) correct?
2
𝑃(𝐵) = 0.4 and P(A ∪ B) = 0.6, then P(A ∩ B) a. 𝑃(𝑋 ∪ 𝑌) = b. X and Y are independent
3
1 2 1 2
a. b. c. d. c. X and Y are not independent d. P(X C ∩ Y) = 3
1
3 3 5 15

1
112. Let 𝐴 and 𝐵 be two events such that 𝑃(𝐴 ∩ 𝐵) = , 117. Let E and F be two independent events. The
6
31 7 11
P(A ∪ B) = ̅) =
and P(B , then probability that exactly one of them occurs is 25 and the
45 10
2
a. 𝐴 and 𝐵 are independent b. A and B are mutually probability of none of them occurring is . If P(T)
25
A 1 B 1
exclusive c. P (B) < 6 d. P (A) < 6 denotes the probability of ccurrence of the event 𝑇,
then
113. A candidate takes three tests in succession and
4 3 1 2
the probability of passing the first test is p. The a. P(E) = , P(F) = b. P(E) = , P(F) =
5 5 5 5
p 2 1 3 4
probability of passing each succeeding test is p or c. P(E) = 5 , P(F) = 5 d. 𝑃(𝐸) = 5 , 𝑃(𝐹) = 5
2

according as he passes or fails in the preceding one. 118. If P(S) = 0.3, P(T) = 0.4. S and T are independent
The candidate is selected if he; passes at least two
events, then P(S ∣ T)
tests. The probability that the candidate is selected is
a. 0.2 b. 0.3 c. 0.12 d. 0.4
a. 𝑝(2 − 𝑝) b. 𝑝 + 𝑝2 + 𝑝 3 c. 𝑝2 (1 − 𝑝) d. p2 (2 − p)
119. A certain item is manufactured by machine M1 and
1
̅̅̅̅̅̅̅
114. Let 𝐴 and 𝐵 be two events such that 𝑃(𝐴 ∪ 𝐵) = 6, M . It is known that machine M turns out twice as
2 1
1 1
̅ ) = , where A
P(A ∩ B) = 4 and P(A ̅ stands for the many items as machine M2 . It is also known that 4% of
4

complement of the event 𝐴. Then the events 𝐴 and 𝐵 the items produced by machine 𝑀1 and 3% of the items

are produced by machine M2 are defective. All the items

a. equally likely but not independent produced are put into one stock pile and then one item

b. independent but not equally likely is selected at random. The probaility that the selected

c. independent and equally likely item is defective is equal to

d. mutually exclusive and independent a. 10/300 b. 11/300 c. 10/200 d. 11/200

Prepared by Anuj Seth Contact 9310913371

 AGNI 4.0
🔥Jab tak todega nahi chodega nahi🔥
DPP – 9 (Probability)

120. A computer producing factory has only two plants respectively. Probability that he reaches office late, if
𝑇1 and T2 . Plant T1 produces 20% and plant T2 he takes car, scooter, bus or train is 2/9,1/9,4/9 and
produces 80% of the total computers produced. 7% of 1/9 respectively. Given that he reached office in time,
computers produced in the factory turn out to be the probability that he travelled by a car is
defective. It is known that P (computer turns out to be a. 1/7 b. 2/7 c. 3/7 d. 4/7
defective given that it is produced in plant T1 ) = 10P( 124. A certain item is manufactured by 3 factory 𝐹1 , 𝐹2
computer turns out to be defective given that its is and 𝐹3 with 30% of item made in F1 , 20% in F2 and 50%
produced in plant 𝑇2 ). A computer produced in the in F3 . It is found that 2% of the items produced by
factory is randomly selected and it does not turn out to 𝐹1 , 3% of the items produced by F2 and 4% of the items
be defective. Then the probability that it is produced in produced by F3 are defective. Suppose that an items
plant 𝑇2 is selected at random from the stock is found defective.
36 47 78 75
a. b. c. d. What is the probability that the item came from F1 ?
73 79 93 83
1 1 1 3
a. 16
b. 8
c. 3
d. 16
121. A bag contains one marble which is either green
or blue, with equal probability. A green marble is put in 125. A student answers a multiple choice question with
the bag (so there are 2 marbles now) and then a 5 alternatives of which exactly one is correct. The
marble is picked at probability that he knows the correct answer is p, 0 <

random from the bag. If the marble taken out is green, p < 1. If he does not know the correct answer, he

then the probability that the remaining marble is also randomly ticks one answer. Given that he has

green is answered the question correctly, the probability that he

a. ½ b. 1 c. 2/3 d. 1/3 did not tick the answer randomly, is

3𝑝 5𝑝 5p 4𝑝
122. A crime is committed by one of the two suspects, a. 4𝑝+3
b. 3𝑝+2
c. 4p+1
d. 3𝑝+1
A and B. Initially, there is equal evidence against both
126. There are two coins, one unbiased with probability
of them. In further investigation at the crime scene, it is 1
of getting heads and the other one is biased with
found that the guilty party has a blood type found in 2
3
20% of the population. If the suspect A does match this probability 4 of getting heads. A coin is selected at
blood type, whereas the blood type of suspect B is random and tossed. It shows heads up. Then the
unknown, then the probability that A is the guilty party probability that the unbiased coin was selected is
is a. 2/3 b. 3/5 c. ½ d. 2/5
a. 3/5 b. 5/6 c. 1/3 d. 2/3
127. A purse contains 4 copper and 3 silver coins, and
123. A person goes to office by car or scooter or bus or
a second purse contains 6 copper and 2 silver coins. A
train, probability of which are 1/7,3/7,2/7 and 1/7
coin is taken out from any purse, the probability that it

Prepared by Anuj Seth Contact 9310913371

 AGNI 4.0
🔥Jab tak todega nahi chodega nahi🔥
DPP – 9 (Probability)

is copper coin is 132. A signal which can be green or red with probability
4 1
a. 3/7 b. 4/7 c. ¾ d. 37/56 and 5 respectively, is received by station A and then
5

128. Two coins are available, one fair and the other transmitted to station B. The probability of each station
two- headed. Choose a coin and toss it once; assume 3
receiving the signal correctly is 4. If the signal received
that
at station B is green, then the probability that the
3
the unbiased coin is chosen with probability 4. Given original signal was green is
3 6 20 9
that the outcome is head, the probability that the two- a. b. c. d.
5 7 23 20

headed coin was chosen is

133. The chances of defective screws in three boxes A,
a. 3/5 b. 2/5 c. 1/5 d. 2/7 1 1 1
B C are 5 , 6 , 7 respectively. A box is selected at random
Directions: Questions 129 and 130 are based on the
and a screw draw from it at random is found to be
following paragraph.
defective. Then, the probability that it came from box A,
Let U1 and U2 be two urns such that U1 contains 3
is
white and 2 red balls, and 𝑈2 contains only 1 white ball. 16 1 27 42
a. 29
b. 15
c. 59
d. 107
A fair coin is tossed. If head appears then 1 ball is
drawn at random from U1 and put into U2 . However, if 134. The probability distribution of X is
tail appears then 2 balls are drawn at random from U1
and put into U2 . Now 1 ball is drawn at random from U2 .
129. The probability of the drawn ball from U2 being
white is The value of k is
13 23 19 11
a. b. c. d. a. 0.7 b. 0.3 c. 1 d. 0.14
30 30 30 30

130. Given that the drawn ball from U2 is white, the 135. For the following distribution function 𝐹(𝑋) of a
probability that head appeared on the coin is r.v.X
17 11 15 12
a. 23
b. 23
c. 23
d. 23

131. Bag I contains 3 red and 4 black balls while

another bag II contains 5 red and 6 black balls. One
ball is drawn at random from one of the bags and it is P(3 < X ≤ 5) =
found to be black. The probability that it was drawn a. 0.48 b. 0.37 c. 0.27 d. 1.47
from bag II is 136. The probability distribution of a random variable is
7 13 21
a. 43
b. 43
c. 43
d. None of these given below:

Prepared by Anuj Seth Contact 9310913371

 AGNI 4.0
🔥Jab tak todega nahi chodega nahi🔥
DPP – 9 (Probability)

142. Probability that a person will develop immunity

after vaccination is 0.8 . If 8 people are given the
vaccine then probability that all develop immunity is
Then P(0 < X < 5) = a. (0.2)8 b. (0.8)8 c. 1 d. 8
C6 (0.2)6 (0.8)2
1 3 8 7
a. 10
b. 10
c. 10
d. 10 143. If the mean and variance of a binomial distribution
137. A random variable X has the probability are 4 and 2 respectively, then the probability of 2
distribution given below. Its variance is successes of that binomial variate X, is
1 219 37 7
a. b. c. d.
2 256 256 64

144. An experiment succeeds twice as often as it fails.

The probability of at least 5 successes in the six trials
4 5 10 16
a. b. c. d. of this experiment is
3 3 3 3
𝑘 496 192 240 256
138. Suppose 𝑓(𝑥) = is a probability distribution of a a. b. c. d.
2𝑥 729 729 729 729

random variable X that can take on the value x = 145. India play two matches each with West Indies and
0,1,2,3, 4. Then k is equal to Australia. In any match, the probabilities of India getting
a. 16/15 b. 15/16 c. 31/16 d. None of these 0,1 and 2 points are 0.45,0.05 and 0.50 respectively.
139. A box contains 15 green and 10 yellow balls. If 10 Assuming that the outcomes are independent, the
balls are randomly drawn, one-by-one, with probability of India getting at least 7 points is
1
replacement, then the variance of the number of green a. 0.0875 b. c. 0.1125 d. None of these
16
balls drawn is 146. The probability that an event does not happen in
6 12
a. 6 b. 4 c. d. one trial is 0.8 . The probability that the event happens
25 5

140. An unbiased coin is tossed eight times. The atmost once in three trials is
probability of obtaining at least one head and at least a. 0.896 b. 0.791 c. 0.642 d. 0.592
one tail is 147. If the mean and variance of a binomial variate 𝑋
63 255 127 1 are 8 and 4 respectively, then P(X < 3) =
a. 64
b. 256
c. 128
d. 2
137 697 265 265
a. 216
b. 216
c. 216
d. 215
141. A box has 100 pens of which 10 are defective.
The probability that out of a sample of 5 pens drawn 148. If the mean and the variance of a binomial variate
one by one with replacement and atmost one is X and 2 and 1 respectively, then the probability that X
defective is takes a value greater than or equal to one is
9 1 9 4 9 5 1 9 4 1 9 5 1 9 3 15
a. b. 2 (10) c. (10) + 2 (10) d. ( ) a. 16
b. 16
c. 4
d. 16
10 5 2 10

Prepared by Anuj Seth Contact 9310913371

 AGNI 4.0
🔥Jab tak todega nahi chodega nahi🔥
DPP – 9 (Probability)

149. If getting a number greater than 4 is a success in 155. A multiple choice examination has 5 questions.
a throw of a fair die, then the probability of at least 2 Each question has three alternative answers of which
successes in six throws of a fair die is exactly one is correct. The probability that a student will
a. 0.649 b. 0.351 c. 0.267 d. 0.667 get 4 or more correct answers just by guessing is
10 17 13 11
150. A coin is tossed 2𝑛 times. The chance that the a. 35
b. 35
c. 35
d. 35

number of times one gets head is not equal to the

156. If mean and variance of a binomial variate X are 2
number of times one gets tail is
and 1 respectively, then the probability that X takes a
(2n)! 1 2n (2n)! (2n)! 1
a. ⋅( ) b. 1 − c. 1 − ⋅ value at least one is
(n!)2 2 (n!)2 (n!)2 4n
1 3 5 15
d. None of these a. b. c. d.
16 16 16 16

151. A man takes a step forward with probability 0.4 157. The probability that an event A happens in one
and one step backwards with probability 0.6 , then the trial of an experiment is 0.4 . If 3 independent trials are
probability that at the end of eleven steps he is one performed, the probability that A happens atleast once
step away from the starting point, is is
11
a. C6 × (0.24)5 b. 11
C6 × (0.72)6 a. 0.936 b. 0.784 c. 0.904 d. None of these
11
c. C5 × (0.48)5 d. 11
C5 × (0.12)5
158. A fair coin is tossed 100 times. The probability of
152. A fair coin is tossed a fixed number of times. If the getting tail an odd number of times is
probability of getting exactly 3 heads equals the 1 1
a. 2
b. 4
c. 0 d. 1
probability of getting exactly 5 heads, then the
159. Consider 5 independent Bernoulli's trials each with
probability of getting exactly one head is
probability of success p. If the probability of at least one
a. 1/64 b. 1/32 c. 1/16 d. 1/8
31
failure is greater than or equal to 32, then p lies in the
153. The mean and variance of a random variable X
interval
having a binomial distribution are 4 and 2 respectively,
1 11 1 3 3 11
find the value of P(X = 1). a. [0, 2] b. (12 , 1] c. ( , ]
2 4
d. ( , ]
4 12
1 1 1 1
a. b. c. d. 160. A fair coin is tossed n number of times. If the
4 16 8 32
probability of having at least one head is more than
154. A box contains 100 bulbs, out of which 10 are
90%, then n is greater than or equal to
defective. A sample of 5 bulbs is drawn. The probability
a. 2 b. 3 c. 4 d. 5
that none is defective is
161. For the binomial distribution (𝑝 + 𝑞)𝐧 , whose mean
9 1 5 9 5 1 5
a. b. ( ) c. ( ) d. ( ) is 20 and variance is 16 , pair (m, p) is
10 10 10 2
1 2 1 2
a. (100, ) b. (100, ) c. (50, ) d. (50, )
5 5 5 5

Prepared by Anuj Seth Contact 9310913371

 AGNI 4.0
🔥Jab tak todega nahi chodega nahi🔥
DPP – 9 (Probability)

162. Two dice are tossed 6 times, then the probability parameter 𝑝 of 𝑋 is
that 7 will show in exactly four of the tosses is 1 1 2 3
a. b. c. d.
3 2 3 4
225 116 125
a. 18442
b. 20003
c. 15552
d. None of these
169. The probability that a certain kind of component
1
163. In a binomial distribution B (n, p − ), if the will survive a given shock test is 4. The probability that
3
4

probability of at least one success is greater than or exactly 2 of the next 4 components tested survive is
9
equal to , then n is greater than a.
9
b.
25
c.
1
d.
27
10
41 128 5 128
1 9 4
a. log b. c. d.
10 4+log10 3 log10 4−log10 3 log10 4−log10 3 170. A random variable X follows binomial distribution
1
log10 4−log10 3
with mean 𝛼 and variance 𝛽. Then

Directions: Questions 164, 165 and 166 are based on a. 0<𝛼<𝛽 b. 0<𝛽<𝛼

the following paragraph. c. 𝛼<0<𝛽 d. 𝛽<0<𝛼

171. A pair of fair dice is thrown independently three
A fair die is tossed repeatedly until a six is obtained. Let
times. The probability of getting a score of exactly 9
X denote the number of tosses required.
twice is
164. The probability that X = 3 equals
a. 8/729 b. 8/243 c. 1/729 d. 8/9
25 25 5 125
a. b. c. d.
216 36 36 216 172. The mean and variance of a random variable 𝑋
165. The probability that X ≥ 3 equals having a binomial distribution are 4 and 2 respectively.
125 25 5 25
a.
216
b.
36
c.
36
d.
216
Then P(X > 6) =

166. The conditional probability that 𝑋 ≥ 6 given 𝑋 > 3 a. 1/256 b. 3/256 c. 9/256 d. 7/256
equals 173. 𝐴 and 𝐵 are two equally strong players. Find the

a.
125
b.
25
c.
5
d.
25 probability that A beats 𝐵 in exactly 3 games out of 4
216 216 36 36
3 1 1 3
a. b. c. d.
167. Two dice are thrown n times in succession. The 4 2 4 7

probability of obtaining a double six at least once is 174. If X follows a binomial distribution with parameters
1 n 35 n 1 n 1
a. (36) b. 1 − (36) c. (12) d. None of these n = 100 and p = 3, then P(X = r) is maximum when r is

168. If 𝑋 is a binomial variate with the range equal to

{0,1,2,3,4,5,6} and 𝑃(𝑋 = 2) = 4𝑃(𝑋 = 4), then the a. 16 b. 32 c. 33 d. None of these

ANSWER KEY: PROBABILITY

1 C 23 A 45 D 67 D 89 B 111 D 133 D 155 D

Prepared by Anuj Seth Contact 9310913371

 AGNI 4.0
🔥Jab tak todega nahi chodega nahi🔥
DPP – 9 (Probability)

2 D 24 C 46 A 68 D 90 C 112 A 134 D 156 D

3 A 25 C 47 C 69 A 91 D 113 D 135 B 157 B

4 B 26 D 48 D 70 D 92 A 114 B 136 C 158 A

5 A 27 C 49 B 71 D 93 C 115 A 137 A 159 A

6 C 28 B 50 C 72 A 94 B 116 C,B 138 D 160 C

7 C 29 D 51 C 73 A 95 D 117 A,D 139 D 161 A

8 A 30 A 52 B 74 C 96 D 118 B 140 C 162 C

9 A 31 C,D 53 C 75 B 97 D 119 B 141 C 163 D

10 C 32 D 54 A 76 D 98 A 120 C 142 B 164 A

11 D 33 A 55 D 77 D 99 A 121 C 143 D 165 B

12 C 34 A 56 D 78 B 100 D 122 B 144 D 166 D

13 B 35 A 57 C 79 C 101 B 123 A 145 A 167 B

14 A 36 A 58 D 80 D 102 C 124 D 146 A 168 A

15 D 37 A 59 A 81 C 103 A 125 C 147 A 169 D

16 B 38 D 60 B 82 A 104 B,C 126 D 148 D 170 B

17 C 39 B 61 D 83 B 105 B 127 D 149 A 171 B

18 C 40 D 62 A 84 C 106 D 128 B 150 C 172 C

19 D 41 A 63 C 85 D 107 B 129 B 151 A 173 C

20 A 42 C 64 A 86 B 108 A 130 D 152 B 174 A

21 B 43 D 65 C 87 C 109 B 131 C 153 D

22 D 44 D 66 A 88 C 110 B 132 C 154 C

Prepared by Anuj Seth Contact 9310913371