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
Correct — B, 3/8. Three tosses give 2 × 2 × 2 = 8 equally likely outcomes. Exactly two heads means one tail, and the tail can fall on the first, second or third toss — HHT, HTH, THH. That is 3 favourable outcomes out of 8, a probability of 3/8. The binomial route gives the same figure: 3C2 × (1/2)² × (1/2)¹ = 3 × 1/8.
- (a)1⁄3 — One-third comes from counting only three outcomes — two heads, three heads, fewer heads — and treating them as equally likely. They are not; the sample space has eight members, not three.
- (c)1⁄2 — A half would need 4 of the 8 outcomes to have exactly two heads. Only 3 do; the fourth head-heavy outcome, HHH, has three heads and does not qualify.
- (d)5⁄8 — 5/8 is the probability of getting at least one tail and at least one head, or of 'two or fewer heads' counted loosely. The word 'exactly' rules out both readings.
For a fair coin every sequence of n tosses is equally likely, each with probability 1/2ⁿ, so a probability question becomes a counting question. The number of sequences with exactly r heads in n tosses is nCr.
The word 'exactly' is doing all the work, and the wrong options are built around candidates who read it as 'at least'. Writing out all eight outcomes takes about fifteen seconds and is the safest method under exam pressure: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. Three of them carry two heads.
- n tosses of a coin give 2ⁿ equally likely outcomes; three tosses give 8.
- P(exactly r heads in n tosses) = nCr / 2ⁿ for a fair coin.
- The counts for three tosses run 1, 3, 3, 1 for zero, one, two and three heads — the fourth row of Pascal's triangle, adding to 8.
- 'Exactly two' and 'at least two' are different events: at least two heads has probability 4/8 = 1/2, since HHH is included.
The counts 1, 3, 3, 1 are the binomial coefficients for n = 3 and always add to 2³ = 8.
- Reading 'exactly two heads' as 'at least two heads', which gives 1/2.
- Counting outcome types rather than outcomes, which gives 1/3.
- Forgetting that the tail can occupy any of the three positions.
Asked as a small coin or dice experiment with a stated number of successes, occasionally with the coin replaced by a biased one.
A fair coin is tossed three times and the outcomes are noted. What is the probability of getting exactly two heads?
- (a) 2/3
- (b) 1/2
- (c) 5/8
- (d) 3/8
Answer(d) 3/8
The identical experiment reset two years later with the options shuffled, which is about as clear a signal as this examiner gives that a question is worth mastering rather than memorising.
- practice — not a real PYQ
A coin is tossed 3 times. The probability of getting at least two heads is
- (a)3/8
- (b)1/2
- (c)5/8
- (d)3/4
Answer(b) 1/2 — the favourable outcomes are HHT, HTH, THH and HHH, that is 4 out of 8.
- practice — not a real PYQ
Two fair dice are rolled together. The probability that the sum is 7 is
- (a)1/12
- (b)1/9
- (c)1/6
- (d)5/36
Answer(c) 1/6 — six of the 36 ordered pairs add to 7, and 6/36 = 1/6.