Six random pieces of a stick all at most half

A stick of length $1$ is cut at $5$ points chosen independently and uniformly along it, producing $6$ pieces. What is the probability that none of the pieces is longer than $\tfrac12$?

$\tfrac{13}{16}$. At most one piece can exceed $\tfrac12$, so the bad events are disjoint. A given piece exceeds $\tfrac12$ with probability $(1-\tfrac12)^{5}=\tfrac1{32}$ (the spacings are exchangeable, so each piece behaves like the first). Hence the answer is $1-6\cdot\tfrac1{32}=\tfrac{13}{16}$.