Breaking a Stick Into Three Pieces That Form a Triangle

Breaking a stick at two random points gives three pieces that form a triangle with probability exactly one quarter, found by geometric probability.

A stick of length 1 is broken at two points chosen independently and uniformly at random along its length, producing three pieces. The probability that those three pieces can be assembled into a triangle is exactly one quarter, a clean geometric result that follows directly from the triangle inequality applied to a uniform distribution over the unit square.

Take a stick of length 1 and choose two break points independently and uniformly at random from the interval between 0 and 1. Breaking the stick at both points produces three pieces. The question asks for the probability that the three resulting lengths satisfy the triangle inequality, meaning they can be arranged edge to edge to form a genuine triangle, as opposed to lying flat or leaving a gap.

Let the two break points be X and Y, drawn independently and uniformly from 0 to 1, and suppose for now X is less than Y, which happens with probability one half and is symmetric to the case Y less than X. Given X less than Y, the three piece lengths are X, Y minus X, and 1 minus Y.

The pair X, Y is uniformly distributed over the unit square, and the condition X less than Y restricts attention to the triangular half of that square above the diagonal, an area of one half. Within that region, every point corresponds to one specific set of three piece lengths, so the probability of forming a triangle can be computed as the area of the sub-region satisfying the triangle inequality, divided by the total area of the region being considered.

Three lengths form a triangle exactly when each length is strictly less than the sum of the other two, which for three pieces summing to a fixed total of 1 is equivalent to the simpler condition that no single piece is one half or longer, since a piece that large could never be spanned by the other two joined together. So the condition becomes: X is less than one half, Y minus X is less than one half, and 1 minus Y is less than one half, all three at once.