Shortest of three random pieces under five centimetres

A one-metre log is cut at two points chosen independently and uniformly along its length, producing three pieces. What is the probability that the smallest piece is $5$ cm or shorter?

$111/400$. All three pieces exceed length $a$ with probability $(1-3a)^2$ (the region where every piece exceeds $a$ is a scaled copy of the full simplex). With $a=0.05$ this is $0.85^2=0.7225$, so the answer is $1-0.7225=\tfrac{111}{400}$.