Probability— Concepts, Formulas & Shortcuts
- P(E) = favourable outcomes / total outcomes; 0 ≤ P(E) ≤ 1.
- P(not E) = 1 − P(E) — use it whenever "at least one" appears.
- Two dice → 36 outcomes; a deck → 52 cards, 26 red, 26 black, 4 aces, 12 face cards.
- Mutually exclusive: P(A or B) = P(A) + P(B). Otherwise subtract P(A and B).
- Independent events: P(A and B) = P(A) × P(B).
- For "drawn together", use combinations: C(n, r) = n!/(r!(n − r)!).
Probability Practice Questions with Answers
Attempt each question first, then open the explanation. All 24 questions below are free to read and require no signup.
Q1.Two dice are thrown together. The probability that the sum is 8 is:
Moderate- A1/9
- B5/36
- C1/6
- D7/36
Probability question 1 of 24+Show Answer & Explanation
Answer: B. 5/36
Explanation: Favourable pairs: (2,6),(3,5),(4,4),(5,3),(6,2) = 5 out of 36 → 5/36.
Q2.A bag contains 4 red and 5 blue balls. One ball is drawn at random. P(red) is:
Easy- A4/9
- B5/9
- C1/2
- D4/5
Probability question 2 of 24+Show Answer & Explanation
Answer: A. 4/9
Explanation: Total = 9 balls; favourable = 4 → 4/9.
Q3.A card is drawn from a well-shuffled pack of 52. The probability that it is a king is:
Easy- A1/26
- B1/13
- C3/13
- D1/4
Probability question 3 of 24+Show Answer & Explanation
Answer: B. 1/13
Explanation: 4 kings out of 52 → 4/52 = 1/13.
Q4.Two coins are tossed. The probability of getting at least one head is:
Easy- A1/4
- B1/2
- C3/4
- D1
Probability question 4 of 24+Show Answer & Explanation
Answer: C. 3/4
Explanation: P(no head) = 1/4, so P(at least one head) = 1 − 1/4 = 3/4.
Q5.A bag has 3 white and 4 black balls. Two balls are drawn at random. The probability that both are black is:
Moderate- A2/7
- B3/7
- C4/7
- D1/3
Probability question 5 of 24+Show Answer & Explanation
Answer: A. 2/7
Explanation: C(4,2)/C(7,2) = 6/21 = 2/7.
Q6.The probability that a card drawn from a pack of 52 is a face card is:
Easy- A1/13
- B3/13
- C4/13
- D1/4
Probability question 6 of 24+Show Answer & Explanation
Answer: B. 3/13
Explanation: 12 face cards (J, Q, K in 4 suits) → 12/52 = 3/13.
Q7.A die is thrown once. The probability of getting a prime number is:
Easy- A1/3
- B1/2
- C2/3
- D5/6
Probability question 7 of 24+Show Answer & Explanation
Answer: B. 1/2
Explanation: Primes on a die are 2, 3 and 5 → 3/6 = 1/2.
Q8.Two dice are rolled. The probability of getting a doublet is:
Easy- A1/12
- B1/6
- C1/4
- D1/3
Probability question 8 of 24+Show Answer & Explanation
Answer: B. 1/6
Explanation: 6 doublets out of 36 outcomes → 6/36 = 1/6.
Q9.A coin is tossed once. The probability of getting a head is:
Easy- A0
- B1/4
- C1/2
- D1
Probability question 9 of 24+Show Answer & Explanation
Answer: C. 1/2
Explanation: One favourable outcome out of two equally likely outcomes → 1/2.
Q10.A die is thrown. The probability of getting a number greater than 4 is:
Easy- A1/6
- B1/3
- C1/2
- D2/3
Probability question 10 of 24+Show Answer & Explanation
Answer: B. 1/3
Explanation: Favourable outcomes are 5 and 6 → 2/6 = 1/3.
Q11.Three coins are tossed. The probability of getting exactly two heads is:
Moderate- A1/8
- B1/4
- C3/8
- D1/2
Probability question 11 of 24+Show Answer & Explanation
Answer: C. 3/8
Explanation: Favourable outcomes HHT, HTH, THH = 3 out of 8 → 3/8.
Q12.A card is drawn from a pack of 52. The probability that it is a red card is:
Easy- A1/4
- B1/3
- C1/2
- D3/4
Probability question 12 of 24+Show Answer & Explanation
Answer: C. 1/2
Explanation: 26 red cards out of 52 → 1/2.
Q13.Two dice are thrown. The probability that the sum is 7 is:
Moderate- A1/9
- B1/6
- C5/36
- D7/36
Probability question 13 of 24+Show Answer & Explanation
Answer: B. 1/6
Explanation: Six favourable pairs out of 36 → 6/36 = 1/6.
Q14.A bag has 5 red, 4 blue and 3 green balls. The probability of drawing a green ball is:
Easy- A1/4
- B1/3
- C3/8
- D5/12
Probability question 14 of 24+Show Answer & Explanation
Answer: A. 1/4
Explanation: Total = 12 balls; favourable = 3 → 3/12 = 1/4.
Q15.If P(E) = 0.35, then P(not E) is:
Easy- A0.35
- B0.55
- C0.65
- D0.75
Probability question 15 of 24+Show Answer & Explanation
Answer: C. 0.65
Explanation: P(not E) = 1 − 0.35 = 0.65.
Q16.A number is chosen at random from 1 to 20. The probability that it is a multiple of 3 is:
Moderate- A1/5
- B3/10
- C1/4
- D1/3
Probability question 16 of 24+Show Answer & Explanation
Answer: B. 3/10
Explanation: Multiples of 3 up to 20: 3, 6, 9, 12, 15, 18 → 6/20 = 3/10.
Q17.A die is thrown. The probability of getting an even number is:
Easy- A1/6
- B1/3
- C1/2
- D2/3
Probability question 17 of 24+Show Answer & Explanation
Answer: C. 1/2
Explanation: Even faces are 2, 4 and 6 → 3/6 = 1/2.
Q18.Two coins are tossed. The probability of getting two tails is:
Easy- A1/4
- B1/3
- C1/2
- D3/4
Probability question 18 of 24+Show Answer & Explanation
Answer: A. 1/4
Explanation: One favourable outcome (TT) out of four → 1/4.
Q19.A card is drawn from 52. The probability that it is a spade is:
Easy- A1/13
- B1/4
- C1/2
- D3/13
Probability question 19 of 24+Show Answer & Explanation
Answer: B. 1/4
Explanation: 13 spades out of 52 → 1/4.
Q20.A bag has 6 white and 4 black balls. The probability of drawing a white ball is:
Easy- A2/5
- B1/2
- C3/5
- D2/3
Probability question 20 of 24+Show Answer & Explanation
Answer: C. 3/5
Explanation: There are 10 balls in all and 6 are white, so the probability is 6/10 = 3/5.
Q21.Two dice are thrown. The probability that the sum is 12 is:
Moderate- A1/36
- B1/18
- C1/12
- D1/6
Probability question 21 of 24+Show Answer & Explanation
Answer: A. 1/36
Explanation: Only (6,6) works → 1/36.
Q22.The probability of an impossible event is:
Easy- A0
- B0.5
- C1
- D−1
Probability question 22 of 24+Show Answer & Explanation
Answer: A. 0
Explanation: An impossible event has no favourable outcome, so its probability is 0.
Q23.A number is chosen from 1 to 10. The probability that it is prime is:
Moderate- A1/5
- B3/10
- C2/5
- D1/2
Probability question 23 of 24+Show Answer & Explanation
Answer: C. 2/5
Explanation: Primes are 2, 3, 5 and 7 → 4/10 = 2/5.
Q24.Three coins are tossed. The probability of getting at least one tail is:
Moderate- A1/8
- B3/8
- C5/8
- D7/8
Probability question 24 of 24+Show Answer & Explanation
Answer: D. 7/8
Explanation: P(no tail) = 1/8, so P(at least one tail) = 1 − 1/8 = 7/8.
Probability — Frequently Asked Questions
When should I use the complement rule?+
Whenever the question says "at least one". P(at least one head in two tosses) = 1 − P(no head) = 1 − 1/4 = 3/4, which is far quicker than listing cases.
Do I use permutations or combinations for drawing balls?+
Combinations, because the order of drawing does not matter. Drawing 2 black balls from 4 in a bag of 7 is C(4,2)/C(7,2) = 6/21 = 2/7.
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