Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter ...

Question

Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is ________ .

Accepted Answer

Expected length of the shorter stick = 0.25. A stick of length 1 is broken at a point U chosen uniformly on [0,1]. The two pieces have lengths U and 1-U. The shorter piece has length min(U, 1-U). We need E[min(U, 1-U)]. By symmetry, if U < 1/2, the shorter piece is U; if U > 1/2, the shorter piece is 1-U. By symmetry both cases contribute equally.