CAPF AC 2022 Q43: General Mental Ability – Quantitative Aptitude
A coin is tossed 3 times. The probability of getting exactly 2 heads is
- (a) 1⁄3
- (b) 3⁄8
- (c) 1⁄2
- (d) 5⁄8
Answer: (b) 3⁄8
Explanation
Exactly two heads in three tosses
Given: a coin is tossed 3 times; find the probability of getting exactly 2 heads.
How to think: list all the outcomes.
Three tosses give 2 × 2 × 2 = 8 equally likely outcomes.
Count those with exactly two heads.
| Number of heads | Outcomes | Count |
|---|---|---|
| 3 | HHH | 1 |
| 2 | HHT, HTH, THH | 3 |
| 1 | HTT, THT, TTH | 3 |
| 0 | TTT | 1 |
Probability: 3 favourable out of 8 = 3/8.
Answer: (b) 3⁄8.
Why each option is right or wrong
- (a) Incorrect. 1/3 assumes only three outcomes; there are eight.
- (b) CORRECT. Three of the eight outcomes have exactly two heads, so the probability is 3/8.
- (c) Incorrect. 1/2 is the chance of at least two heads (4 out of 8).
- (d) Incorrect. 5/8 has no matching count of outcomes here.
Key point
n tosses give 2ⁿ outcomes. Exactly r heads: nCr ways. For 3 tosses: 1, 3, 3, 1 (Pascal's triangle). Exactly 2 heads = 3/8; at least 2 heads = 4/8 = 1/2.
Remember
Three tosses: 1, 3, 3, 1 out of 8.
Common mistakes
- Counting HHH as well and answering 4/8, which is at least two heads.
- Thinking there are only 4 outcomes (0, 1, 2 or 3 heads) and answering 1/4.
- Forgetting that HHT, HTH and THH are different outcomes.
Exam tip
Educated guess: the bottom number must be 8, since three tosses give eight outcomes. Only (b) and (d) qualify, and you cannot find five ways to get two heads.
Sources
- NCERT Class X Mathematics — Probability
- NCERT Class XI Mathematics — Probability