If you break a unit stick randomly at two points, what is the expected length of the smallest piece?

Free worked solution · Quant Blueprint

1/9, when the two break points are independent and uniform along the original unit stick.

1. Let the ordered cut positions be U < V. The lengths are U, V − U and 1 − V. Set M to the minimum of those lengths.
2. For M > t, all three lengths must exceed t. In the ordered (U, V) triangle, the remaining feasible triangle has side length 1 − 3t, compared with side length one originally.
3. The uniform density makes the probability equal to the area ratio: P(M > t) = (1 − 3t)² for 0 ≤ t ≤ 1/3.
4. Use the tail integral for a nonnegative random variable: E(M) = ∫₀¹⁄³ (1 − 3t)² dt = 1/9.

Common mistake

Cutting once, choosing a resulting piece at random, and cutting that piece is a different random experiment. State how the break points are generated.

Follow-up

The probability the shortest piece exceeds 0.1 is (1 − 0.3)² = 0.49.

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