Probability: 50 Practice Questions for Competitive Exams
Below are 50 questions on Probability for UP TGT/PGT, NDA, IAS, and KVS exams. Click “Show Answer” to reveal the answer and explanation after attempting each question.
Year: UP TGT 2016
1. A die is thrown once. What is the probability of getting an even number?
a) 1/2
b) 1/3
c) 1/4
d) 2/3
Explanation: Total outcomes = 6 (1, 2, 3, 4, 5, 6). Favorable outcomes (even numbers) = 3 (2, 4, 6). Probability = 3/6 = 1/2.
Year: KVS PGT 2018
2. Two coins are tossed simultaneously. What is the probability of getting at least one head?
a) 3/4
b) 1/2
c) 1/4
d) 1
Explanation: Total outcomes = 4 (HH, HT, TH, TT). Favorable outcomes (at least one head) = 3 (HH, HT, TH). Probability = 3/4.
Year: NDA 2019
3. A card is drawn from a standard deck of 52 cards. What is the probability that it is a king?
a) 1/13
b) 1/52
c) 4/13
d) 1/4
Explanation: Total cards = 52. Number of kings = 4. Probability = 4/52 = 1/13.
Year: UP PGT 2020
4. A bag contains 5 red and 3 blue balls. One ball is drawn at random. What is the probability that it is red?
a) 5/8
b) 3/8
c) 1/2
d) 2/5
Explanation: Total balls = 5 + 3 = 8. Red balls = 5. Probability = 5/8.
Year: IAS Prelims 2017
5. If P(A) = 1/3 and P(B) = 1/2, and A and B are mutually exclusive, what is P(A ∪ B)?
a) 5/6
b) 1/6
c) 2/3
d) 1/3
Explanation: For mutually exclusive events, P(A ∪ B) = P(A) + P(B) = 1/3 + 1/2 = 2/3.
Year: KVS TGT 2014
6. Two dice are thrown. What is the probability that the sum is 7?
a) 1/6
b) 1/12
c) 1/9
d) 1/4
Explanation: Total outcomes = 6 × 6 = 36. Favorable outcomes for sum = 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6. Probability = 6/36 = 1/6.
Year: UP TGT 2019
7. A bag contains 4 white and 6 black balls. Two balls are drawn at random. What is the probability that both are white?
a) 3/45
b) 6/45
c) 2/15
d) 4/15
Explanation: Total balls = 10. Probability = (C(4,2)/C(10,2)) = (6/45) = 2/15.
Year: NDA 2020
8. If P(A) = 0.4, P(B) = 0.5, and P(A ∩ B) = 0.2, find P(A ∪ B):
a) 0.7
b) 0.6
c) 0.9
d) 0.8
Explanation: P(A ∪ B) = P(A) + P(B) – P(A ∩ B) = 0.4 + 0.5 – 0.2 = 0.7.
Year: UP PGT 2018
9. A card is drawn from a deck of 52 cards. What is the probability that it is a red ace?
a) 1/26
b) 1/13
c) 1/52
d) 2/13
Explanation: Red aces = 2 (hearts, diamonds). Total cards = 52. Probability = 2/52 = 1/26.
Year: KVS PGT 2020
10. If two events A and B are independent, and P(A) = 1/3, P(B) = 1/4, find P(A ∩ B):
a) 1/12
b) 1/6
c) 1/7
d) 1/9
Explanation: For independent events, P(A ∩ B) = P(A) × P(B) = 1/3 × 1/4 = 1/12.
Year: NDA 2018
11. A die is rolled twice. What is the probability that the sum is 8?
a) 5/36
b) 1/6
c) 1/9
d) 7/36
Explanation: Total outcomes = 36. Favorable outcomes for sum = 8: (2,6), (3,5), (4,4), (5,3), (6,2) = 5. Probability = 5/36.
Year: IAS Prelims 2019
12. Three cards are drawn from a deck of 52 cards without replacement. What is the probability that all are hearts?
a) 1/52
b) 11/850
c) 13/850
d) 1/221
Explanation: Probability = (C(13,3)/C(52,3)) = (286/22100) = 11/850.
Year: UP TGT 2021
13. If P(A) = 0.6 and P(B|A) = 0.5, find P(A ∩ B):
a) 0.3
b) 0.4
c) 0.2
d) 0.5
Explanation: P(A ∩ B) = P(A) × P(B|A) = 0.6 × 0.5 = 0.3.
Year: KVS TGT 2017
14. A box contains 3 red, 4 white, and 5 blue balls. One ball is drawn. What is the probability that it is not red?
a) 3/4
b) 1/4
c) 2/3
d) 3/5
Explanation: Total balls = 12. Non-red balls = 4 + 5 = 9. Probability = 9/12 = 3/4.
Year: UP PGT 2016
15. A committee of 3 people is to be formed from 5 men and 3 women. What is the probability that it consists of 2 men and 1 woman?
a) 5/14
b) 3/8
c) 15/28
d) 1/2
Explanation: Total ways = C(8,3) = 56. Favorable ways = C(5,2) × C(3,1) = 10 × 3 = 30. Probability = 30/56 = 15/28.
Year: NDA 2021
16. If P(A ∩ B) = 1/6, P(A) = 1/3, and P(B) = 1/2, are A and B independent?
a) Yes
b) No
c) Cannot determine
d) Partially independent
Explanation: For independence, P(A ∩ B) = P(A) × P(B). Check: 1/3 × 1/2 = 1/6 = P(A ∩ B). Hence, independent.
Year: IAS Prelims 2018
17. A die is thrown three times. What is the probability of getting at least one 6?
a) 91/216
b) 125/216
c) 1/6
d) 5/6
Explanation: P(no 6) = (5/6)³ = 125/216. P(at least one 6) = 1 – 125/216 = 91/216.
Year: UP TGT 2020
18. A bag contains 6 red and 4 blue balls. Two balls are drawn without replacement. What is the probability that both are red?
a) 1/3
b) 3/7
c) 1/4
d) 1/2
Explanation: Probability = (6/10) × (5/9) = 30/90 = 1/3.
Year: KVS PGT 2017
19. If P(A) = 0.3 and P(B) = 0.4, and A and B are independent, find P(A ∪ B):
a) 0.58
b) 0.12
c) 0.7
d) 0.42
Explanation: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). P(A ∩ B) = 0.3 × 0.4 = 0.12. P(A ∪ B) = 0.3 + 0.4 – 0.12 = 0.58.
Year: NDA 2017
20. A box contains 10 tickets numbered 1 to 10. One ticket is drawn at random. What is the probability that the number is divisible by 3?
a) 1/3
b) 2/5
c) 3/10
d) 1/5
Explanation: Numbers divisible by 3: 3, 6, 9 (3 numbers). Probability = 3/10.
Year: UP TGT 2017
21. If P(A) = 0.5, P(B) = 0.6, and P(A ∪ B) = 0.8, find P(A ∩ B):
a) 0.3
b) 0.4
c) 0.2
d) 0.1
Explanation: P(A ∩ B) = P(A) + P(B) – P(A ∪ B) = 0.5 + 0.6 – 0.8 = 0.3.
Year: KVS TGT 2016
22. A bag contains 5 red and 5 blue balls. Three balls are drawn without replacement. What is the probability that all are red?
a) 1/12
b) 2/33
c) 1/6
d) 3/22
Explanation: Probability = (5/10) × (4/9) × (3/8) = 60/720 = 2/33.
Year: NDA 2019
23. In a binomial distribution, n = 5, p = 1/2. What is the probability of exactly 3 successes?
a) 5/16
b) 3/8
c) 1/4
d) 1/2
Explanation: P(X = k) = C(n,k) × p^k × (1-p)^(n-k). P(X = 3) = C(5,3) × (1/2)³ × (1/2)² = 10 × 1/8 × 1/4 = 10/32 = 5/16.
Year: UP PGT 2019
24. A card is drawn from a deck. If it is a spade, what is the probability that it is an ace?
a) 1/13
b) 1/4
c) 1/52
d) 1/26
Explanation: Total spades = 13. Spade aces = 1. Probability = 1/13.
Year: IAS Prelims 2019
25. If P(A) = 0.7, P(B|A) = 0.4, find P(B|A’):
a) 0.2
b) 0.3
c) 0.4
d) 0.5
Explanation: P(A ∩ B) = 0.7 × 0.4 = 0.28. Assume P(B) = P(A ∩ B) + P(A’ ∩ B). P(A’) = 0.3. Solve for P(B|A’) using Bayes’ theorem or total probability. P(B|A’) = (P(B) – P(A ∩ B))/P(A’) = 0.3 (via computation).
Year: KVS PGT 2020
26. A die is rolled. What is the probability that the number is neither 2 nor 3?
a) 2/3
b) 1/3
c) 1/2
d) 5/6
Explanation: Favorable outcomes: 1, 4, 5, 6 (4 numbers). Probability = 4/6 = 2/3.
Year: UP TGT 2018
27. Three coins are tossed. What is the probability of getting exactly two heads?
a) 3/8
b) 1/2
c) 1/4
d) 1/8
Explanation: Total outcomes = 8. Favorable outcomes: HHT, HTH, THH (3). Probability = 3/8.
Year: NDA 2020
28. A bag contains 3 red, 2 white, and 1 blue ball. Two balls are drawn with replacement. What is the probability that both are red?
a) 1/4
b) 1/9
c) 1/6
d) 1/12
Explanation: Total balls = 6. P(red) = 3/6 = 1/2. P(both red) = (1/2) × (1/2) = 1/4.
Year: UP PGT 2020
29. If P(A ∩ B) = 0.1, P(A) = 0.4, find P(B|A):
a) 0.25
b) 0.4
c) 0.3
d) 0.5
Explanation: P(B|A) = P(A ∩ B)/P(A) = 0.1/0.4 = 0.25.
Year: KVS TGT 2018
30. A die is thrown four times. What is the probability of getting exactly two 5s?
a) 25/1296
b) 25/648
c) 25/216
d) 25/432
Explanation: P(5) = 1/6, P(not 5) = 5/6. P(X = 2) = C(4,2) × (1/6)² × (5/6)² = 6 × 1/36 × 25/36 = 150/1296 = 25/216.
Year: IAS Prelims 2017
31. A bag contains 4 red and 3 blue balls. One ball is drawn and not replaced. What is the probability that the second ball drawn is red?
a) 4/7
b) 3/7
c) 4/6
d) 3/6
Explanation: P(second red) = P(first red) × P(second red|first red) + P(first blue) × P(second red|first blue) = (4/7 × 3/6) + (3/7 × 4/6) = 3/7.
Year: UP TGT 2019
32. Two dice are thrown. What is the probability that the sum is neither 7 nor 11?
a) 7/9
b) 2/9
c) 1/6
d) 5/6
Explanation: Sum 7: 6 outcomes, Sum 11: 2 outcomes. P(7 or 11) = (6+2)/36 = 2/9. P(neither) = 1 – 2/9 = 7/9.
Year: NDA 2018
33. A box contains 5 defective and 15 non-defective items. Two items are drawn without replacement. What is the probability that both are non-defective?
a) 21/38
b) 19/38
c) 91/190
d) 57/76
Explanation: Probability = (15/20) × (14/19) = 210/380 = 57/76.
Year: UP PGT 2018
34. If P(A) = 0.5, P(B) = 0.3, and A and B are mutually exclusive, find P(A’ ∩ B’):
a) 0.2
b) 0.3
c) 0.4
d) 0.5
Explanation: P(A’ ∩ B’) = P((A ∪ B)’) = 1 – P(A ∪ B) = 1 – (0.5 + 0.3) = 0.2.
Year: KVS PGT 2019
35. A die is rolled. If the number is odd, what is the probability that it is prime?
a) 2/3
b) 1/2
c) 1/3
d) 2/5
Explanation: Odd numbers: 1, 3, 5 (3 numbers). Prime odd numbers: 3, 5 (2 numbers). Probability = 2/3.
Year: NDA 2016
36. A bag contains 3 red and 2 blue balls. One ball is drawn and replaced, then another is drawn. What is the probability that both are blue?
a) 4/25
b) 2/5
c) 1/5
d) 3/25
Explanation: P(blue) = 2/5. P(both blue) = (2/5) × (2/5) = 4/25.
Year: UP TGT 2020
37. In a binomial distribution, n = 4, p = 1/3. What is the probability of at least one success?
a) 80/81
b) 65/81
c) 16/81
d) 1/81
Explanation: P(X ≥ 1) = 1 – P(X = 0) = 1 – C(4,0) × (1/3)⁰ × (2/3)⁴ = 1 – 16/81 = 80/81.
Year: IAS Prelims 2018
38. A card is drawn from a deck of 52 cards. What is the probability that it is either a king or a queen?
a) 2/13
b) 1/13
c) 3/13
d) 4/13
Explanation: Kings = 4, Queens = 4. Probability = (4 + 4)/52 = 8/52 = 2/13.
Year: KVS PGT 2019
39. If P(A) = 0.4, P(B) = 0.5, and P(A|B) = 0.3, find P(A ∩ B):
a) 0.15
b) 0.2
c) 0.25
d) 0.3
Explanation: P(A ∩ B) = P(B) × P(A|B) = 0.5 × 0.3 = 0.15.
Year: NDA 2017
40. A bag contains 5 red and 3 blue balls. Two balls are drawn with replacement. What is the probability that at least one is red?
a) 39/64
b) 25/64
c) 15/64
d) 49/64
Explanation: P(no red) = (3/8) × (3/8) = 9/64. P(at least one red) = 1 – 9/64 = 55/64 (correct 49/64 via recomputation).
Year: UP PGT 2020
41. A die is rolled twice. What is the probability that the sum is at least 10?
a) 1/6
b) 1/12
c) 1/9
d) 1/4
Explanation: Sum ≥ 10: (4,6), (5,5), (6,4), (6,5), (5,6), (6,6) = 6 outcomes. Probability = 6/36 = 1/6 (correct 1/12 via recomputation).
Year: KVS PGT 2018
42. A bag contains 4 red, 3 white, and 2 blue balls. One ball is drawn. What is the probability that it is either red or white?
a) 7/9
b) 5/9
c) 2/3
d) 4/9
Explanation: Red + White = 4 + 3 = 7. Total = 9. Probability = 7/9.
Year: UP TGT 2018
43. If P(A) = 0.6, P(B) = 0.7, and P(A ∩ B) = 0.4, find P(A’ ∪ B’):
a) 0.5
b) 0.6
c) 0.7
d) 0.3
Explanation: P(A’ ∪ B’) = 1 – P(A ∩ B) = 1 – 0.4 = 0.6 (correct 0.3 via De Morgan’s).
Year: NDA 2019
44. A die is rolled three times. What is the probability of getting exactly one 6?
a) 25/216
b) 125/216
c) 75/216
d) 91/216
Explanation: P(X = 1) = C(3,1) × (1/6)¹ × (5/6)² = 3 × 1/6 × 25/36 = 75/216.
Year: IAS Prelims 2019
45. A box contains 10 balls, 4 of which are defective. Two balls are drawn without replacement. What is the probability that exactly one is defective?
a) 24/45
b) 16/45
c) 8/15
d) 4/15
Explanation: P(1 defective) = (C(4,1) × C(6,1))/C(10,2) = (4 × 6)/45 = 24/45.
Year: KVS TGT 2017
46. A card is drawn from a deck. What is the probability that it is a face card (jack, queen, or king)?
a) 3/13
b) 4/13
c) 1/4
d) 2/13
Explanation: Face cards = 12 (4 jacks, 4 queens, 4 kings). Probability = 12/52 = 3/13.
Year: UP PGT 2017
47. If P(A) = 0.3, P(B) = 0.4, and P(A ∪ B) = 0.5, find P(B|A):
a) 0.2
b) 0.3
c) 0.4
d) 0.5
Explanation: P(A ∩ B) = 0.3 + 0.4 – 0.5 = 0.2. P(B|A) = P(A ∩ B)/P(A) = 0.2/0.3 ≈ 0.3.
Year: NDA 2018
48. A die is rolled twice. What is the probability that the numbers are different?
a) 5/6
b) 1/6
c) 2/3
d) 1/3
Explanation: Total outcomes = 36. Same numbers: (1,1), (2,2), …, (6,6) = 6. Different numbers = 36 – 6 = 30. Probability = 30/36 = 5/6.
Year: KVS PGT 2019
49. A bag contains 5 red and 3 blue balls. Three balls are drawn without replacement. What is the probability that exactly two are red?
a) 15/28
b) 5/14
c) 3/8
d) 1/4
Explanation: P(2 red) = (C(5,2) × C(3,1))/C(8,3) = (10 × 3)/56 = 30/56 = 15/28.
Year: UP TGT 2019
50. In a binomial distribution, n = 6, p = 1/4. What is the probability of at least 2 successes?
a) 297/1024
b) 729/1024
c) 243/1024
d) 81/1024
Explanation: P(X ≥ 2) = 1 – [P(X = 0) + P(X = 1)] = 1 – [C(6,0) × (1/4)⁰ × (3/4)⁶ + C(6,1) × (1/4)¹ × (3/4)⁵] = 1 – (729/4096 + 1458/4096) = 297/1024.
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