2026-03-23

Problem — 2026-03-23

Medium

Mediumprobability

A stick of length $1$ is broken at two points chosen uniformly at random. What is the probability that the three pieces can form a triangle?

Express as a fraction

Show solution

Let the break points be

X

and

Y

, uniform on

[0,1]

. The three pieces form a triangle iff no piece exceeds

\frac{1}{2}

. In the unit square of

(X,Y)

, the region satisfying all three constraints has area

\frac{1}{4}

(verified by drawing the three inequalities). So the probability is

\frac{1}{4}