Here, we are taking up an important topic: probability. For this topic, we have divided the questions into 3 sections with 15 questions and answers for each. You’ll solve problems on coins, dice, cards, and mixed events. Each question includes an answer for quick checking. Do this series, and probability will feel easy and doable.
Do check out other topics aptitude question and answers.
Probability: Defined Symbols + Core Formulas
- Symbol meanings (first occurrence definitions)
- S = sample space (set of all possible outcomes)
- n(S) = number of outcomes in S
- A, B = events (subsets of S)
- n(A) = number of outcomes in event A
- P(A) = probability of event A
- Aᶜ = complement of A (A does not occur)
- A ∪ B = union (A or B or both)
- A ∩ B = intersection (A and B)
- P(A|B) = probability of A given B has occurred
- p = probability of success in one trial, q = 1 − p
- Equally likely outcomes
- P(A) = n(A) ÷ n(S)
- Basic properties
- 0 ≤ P(A) ≤ 1
- P(S) = 1
- P(∅) = 0
- P(Aᶜ) = 1 − P(A)
- Addition rule
- P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- Mutually exclusive events
- A and B mutually exclusive ⇒ A ∩ B = ∅ ⇒ P(A ∩ B) = 0
- So, P(A ∪ B) = P(A) + P(B)
- Conditional probability
- P(A|B) = P(A ∩ B) ÷ P(B), where P(B) > 0
- Multiplication rule
- P(A ∩ B) = P(A|B) × P(B)
- P(A ∩ B) = P(B|A) × P(A)
- Independent events
- A and B independent ⇔ P(A|B) = P(A) (and P(B|A) = P(B))
- If independent: P(A ∩ B) = P(A) × P(B)
- Bayes’ theorem
- P(A|B) = [P(B|A) × P(A)] ÷ P(B)
- Law of total probability
- If B₁, B₂, …, Bₖ partition S, then
- P(A) = Σ P(A|Bᵢ) × P(Bᵢ)
- Counting notation used in probability (common)
- n! = factorial of n
- C(n,r) = ⁿCᵣ = n! ÷ (r! × (n − r)!)
- P(n,r) = ⁿPᵣ = n! ÷ (n − r)!
- Binomial probability (n independent trials)
- P(X = r) = C(n,r) × pʳ × qⁿ⁻ʳ
- Mean = np
- Variance = npq
- Geometric probability (first success on k-th trial)
- P(X = k) = qᵏ⁻¹ × p
- Mean = 1 ÷ p
- Variance = q ÷ p²
- Hypergeometric probability (without replacement)
- Population size = N, success items = K, draws = n, successes drawn = x
- P(X = x) = [C(K, x) × C(N − K, n − x)] ÷ C(N, n)
- Expectation and variance (discrete)
- E[X] = Σ x × P(X = x)
- Var(X) = E[X²] − (E[X])²
- E[aX + b] = aE[X] + b
- Var(aX + b) = a²Var(X)
45 Probability Aptitude Questions and Answers (Solved MCQs)
Question 1. Three coins and one die are tossed. What is the probability of getting exactly 2 heads and an odd number on the die?
a) 3/32
b) 3/16
c) 5/32
d) 1/8
Answer:
b) 3/16 — 2H from 3 coins and odd on die.
Question 2. Two cards are drawn from a standard deck without replacement. Probability that both are face cards (J, Q, K)?
a) 1/17
b) 11/221
c) 3/52
d) 2/17
Answer:
b) 11/221
Question 3. Three cards are drawn without replacement. Probability of getting exactly 2 aces?
a) 1/221
b) 72/5525
c) 6/221
d) 12/425
Answer:
b) 72/5525
Question 4. Four dice are thrown. Probability of getting at least one 1 and at least one 6?
a) 25/216
b) 151/648
c) 5/36
d) 7/54
Answer:
b) 151/648 — Use inclusion–exclusion on “no 1” and “no 6”.
Question 5. A random 5-digit number (10000–99999) is chosen. Probability that all digits are distinct?
a) 189/625
b) 7/25
c) 4/15
d) 9/25
Answer:
a) 189/625
Question 6. A bag has 5 red, 3 blue, and 2 green balls. Two are drawn without replacement. Probability that they are of different colors?
a) 14/45
b) 31/45
c) 2/3
d) 3/5
Answer:
b) 31/45
Question 7. Three dice are thrown. Probability of getting exactly one pair (two same, third different)?
a) 5/18
b) 1/3
c) 5/12
d) 7/18
Answer:
c) 5/12 — Count pair value, odd value, odd position.
Question 8. Four coins are tossed. Probability of getting at most 1 tail?
a) 5/16
b) 3/8
c) 1/4
d) 7/16
Answer:
a) 5/16
Question 9. One card is drawn. Probability that it is red or a king?
a) 1/2
b) 7/13
c) 9/26
d) 2/13
Answer:
b) 7/13 — Add red and kings, subtract red kings.
Question 10. A die is rolled until the first 6 appears. Probability that it takes more than 3 rolls?
a) 1/6
b) 25/216
c) 125/216
d) 5/6
Answer:
c) 125/216 — No 6 in first 3 rolls.
Question 11. Two dice are thrown. Probability that their product is divisible by 6?
a) 1/2
b) 5/12
c) 7/18
d) 2/3
Answer:
b) 5/12
Question 12. Three cards are drawn without replacement. Probability that all are from the same suit?
a) 1/17
b) 22/425
c) 13/221
d) 3/34
Answer:
b) 22/425
Question 13. A 3-digit number is formed from digits 0–9 without repetition. Probability that it is divisible by 5?
a) 1/9
b) 17/81
c) 2/9
d) 5/27
Answer:
b) 17/81 — Last digit must be 0 or 5.
Question 14. From 6 men and 4 women, two people are chosen randomly. Probability that at least one is a woman?
a) 1/3
b) 2/3
c) 3/5
d) 7/15
Answer:
b) 2/3 — Use complement: both men.
Question 15. Three dice are thrown. Probability that the sum is 10 or 11?
a) 1/6
b) 1/4
c) 5/18
d) 7/27
Answer:
b) 1/4
Question 1. Two cards are drawn without replacement. Given the first card is red, what is the probability that the second card is also red?
a) 1/2
b) 25/51
c) 13/26
d) 12/25
Answer:
b) 25/51
Question 2. Two dice are thrown. Given that the sum is 7, what is the probability that at least one die shows 3?
a) 1/6
b) 1/3
c) 1/2
d) 2/3
Answer:
b) 1/3 — 2 favourable out of 6 sum-7 outcomes.
Question 3. Box A has 3 white and 2 black balls. Box B has 1 white and 4 black balls. A box is chosen at random and a white ball is drawn. What is the probability it came from Box A?
a) 3/5
b) 2/3
c) 3/4
d) 4/5
Answer:
c) 3/4 — Bayes using P(W|A)=3/5, P(W|B)=1/5.
Question 4. One coin is fair and the other is double-headed. A coin is chosen at random and tossed. If the result is head, what is the probability the chosen coin is double-headed?
a) 1/2
b) 2/3
c) 3/4
d) 4/5
Answer:
b) 2/3 — Head is more likely from double-headed coin.
Question 5. A family has two children. Given at least one child is a boy, what is the probability that both are boys?
a) 1/2
b) 1/3
c) 2/3
d) 3/4
Answer:
b) 1/3
Question 6. A family has two children. Given the older child is a boy, what is the probability that both are boys?
a) 1/3
b) 1/2
c) 2/3
d) 3/4
Answer:
b) 1/2
Question 7. A bag contains 5 red and 5 blue balls. Two balls are drawn without replacement. Given the first ball is red, what is the probability the second ball is blue?
a) 4/9
b) 5/9
c) 1/2
d) 2/3
Answer:
b) 5/9
Question 8. A die is rolled twice. Given that at least one 6 appears, what is the probability that exactly one 6 appears?
a) 5/11
b) 10/11
c) 6/11
d) 1/2
Answer:
b) 10/11 — Exactly one 6 divided by at least one 6.
Question 9. In a class, 60% pass Mathematics, 50% pass Science, and 30% pass both. If a student passed Mathematics, what is the probability they also passed Science?
a) 1/3
b) 1/2
c) 2/3
d) 3/5
Answer:
b) 1/2 — P(S|M)=P(S∩M)/P(M).
Question 10. Three boxes have defective rates 2%, 5%, and 10%. A box is chosen with probabilities 0.5, 0.3, 0.2 respectively. An item is found defective. Probability it came from the 10% box?
a) 2/9
b) 1/3
c) 4/9
d) 5/9
Answer:
c) 4/9 — Bayes: 0.02 / 0.045.
Question 11. Two dice are thrown. Given their product is even, what is the probability that both dice show even numbers?
a) 1/2
b) 1/3
c) 2/3
d) 3/4
Answer:
b) 1/3
Question 12. Two cards are drawn without replacement. Given that at least one card is an ace, what is the probability that both are aces?
a) 1/17
b) 1/33
c) 1/52
d) 2/51
Answer:
b) 1/33 — (both aces) ÷ (at least one ace).
Question 13. Three coins are tossed. Given that at least two heads appear, what is the probability that exactly two heads appear?
a) 1/2
b) 2/3
c) 3/4
d) 4/5
Answer:
c) 3/4 — Outcomes with ≥2H are {2H,3H}.
Question 14. Machine A makes 60% items with 1% defective. Machine B makes 40% items with 3% defective. If an item is defective, probability it came from Machine B?
a) 1/3
b) 1/2
c) 2/3
d) 3/4
Answer:
c) 2/3 — Defects weigh B more heavily.
Question 15. Two dice are thrown. Given the first die shows a number greater than the second, what is the probability that the sum is 8?
a) 1/9
b) 2/15
c) 1/6
d) 1/5
Answer:
b) 2/15 — Conditional sample space has 15 outcomes.
Question 1. A random 4-digit number (1000–9999) is chosen. Probability that it is divisible by 9?
a) 1/8
b) 1/9
c) 1/10
d) 2/9
Answer:
b) 1/9
Question 2. A 3-digit number is formed using digits 1–7 without repetition. Probability that the number is even?
a) 2/7
b) 3/7
c) 4/7
d) 1/2
Answer:
b) 3/7 — Last digit must be 2,4,6.
Question 3. A committee of 4 is chosen from 6 men and 5 women. Probability that it has exactly 2 women?
a) 4/11
b) 5/11
c) 6/11
d) 7/11
Answer:
b) 5/11
Question 4. An urn has 4 defective and 6 good items. Three are drawn without replacement. Probability of at most 1 defective?
a) 1/2
b) 2/3
c) 3/4
d) 5/6
Answer:
b) 2/3
Question 5. Ten distinct letters: 5 vowels and 5 consonants are arranged randomly. Probability that all vowels are together?
a) 1/21
b) 1/35
c) 1/42
d) 1/84
Answer:
c) 1/42 — Treat vowels as one block.
Question 6. A number is chosen uniformly from 1 to 100. Probability it is divisible by 2 or 5?
a) 1/2
b) 3/5
c) 2/3
d) 7/10
Answer:
b) 3/5 — Use inclusion–exclusion for multiples.
Question 7. Five coins are tossed. Probability that the number of heads is a prime number?
a) 5/16
b) 21/32
c) 11/16
d) 3/8
Answer:
b) 21/32 — Prime heads: 2,3,5.
Question 8. Two dice are thrown. Probability that their sum is a prime number?
a) 5/12
b) 7/18
c) 1/2
d) 2/3
Answer:
a) 5/12
Question 9. A 2-digit number (10–99) is chosen. Probability that the sum of its digits is 9?
a) 1/9
b) 1/10
c) 2/9
d) 1/12
Answer:
b) 1/10 — There are 9 such two-digit numbers.
Question 10. Five cards are drawn from a standard deck. Probability that the hand contains at least one card from each suit?
a) 2197/8330
b) 1/4
c) 5/13
d) 11/52
Answer:
a) 2197/8330 — Distribution must be 2-1-1-1 across suits.
Question 11. A random 3-digit number (100–999) is chosen. Probability that it is divisible by 4?
a) 1/3
b) 1/4
c) 2/9
d) 3/10
Answer:
b) 1/4
Question 12. From 5 engineers, 4 doctors, and 3 lawyers, 3 people are chosen. Probability that the selection has one from each profession?
a) 2/11
b) 3/11
c) 5/22
d) 1/6
Answer:
b) 3/11
Question 13. A number is chosen uniformly from 1 to 20. Probability that it has exactly two distinct prime factors?
a) 1/4
b) 7/20
c) 2/5
d) 9/20
Answer:
b) 7/20 — Numbers: 6,10,12,14,15,18,20.
Question 14. Digits 1–6 are arranged randomly to form a 6-digit number. Probability that it is divisible by 5?
a) 1/5
b) 1/6
c) 1/10
d) 1/3
Answer:
b) 1/6 — Last digit must be 5.
Question 15. An urn has 3 red, 3 blue, and 3 green balls. Three balls are drawn without replacement. Probability that all three are of different colors?
a) 1/3
b) 9/28
c) 3/14
d) 2/7
Answer:
b) 9/28 — Choose 1 from each color.
