Bear fishing until three catches, fifth attempt

A bear makes a sequence of attempts to catch a fish; each attempt succeeds with probability $\tfrac12$, independently. Once it has caught $3$ fish it stops fishing. What is the probability that the bear makes a fifth attempt and that attempt is successful?

$11/32$. A fifth attempt happens only if at most $2$ of the first $4$ succeeded, which has probability $\tfrac{1+4+6}{16}=\tfrac{11}{16}$. That attempt then succeeds with probability $\tfrac12$, giving $\tfrac{11}{32}$.