What is the probability of obtaining 3 heads in a row when tossing an unbiased coin 5 times?

Free worked solution · Quant Blueprint

1/4, counting at least one run of three or more heads.

1. Let A₁, A₂ and A₃ mean HHH starts at flip one, two or three. Each has probability 1/8.
2. Their pairwise intersections have probabilities 1/16, 1/16 and 1/32: adjacent starts require four heads, while starts one and three require five.
3. All three occur only in HHHHH, with probability 1/32. Inclusion-exclusion gives 3/8 − (1/16+1/16+1/32) + 1/32 = 1/4.

Common mistake

Adding the three start probabilities counts longer runs more than once. Clarify whether the interviewer means at least three consecutive heads or a maximal run of exactly three.

Follow-up

For a maximal run of exactly three and no longer run, only five of the 32 sequences qualify, giving 5/32. The standard at-least-three interpretation includes eight sequences.

Original educational explanation for this practice question. Reviewed 2026-10-13. Firm-style practice; the label describes the collection’s preparation context.