Chance a bear misses the fifth fish

A bear stands by a river and will leave as soon as it has caught 3 fish. Each fish swimming past is caught with probability $\tfrac12$, independently, provided the bear is still there. What is the probability that the 5th fish to swim past is not caught?

$21/32$. The bear is still fishing at fish 5 only if it caught at most 2 of the first 4, probability $\tfrac{1+4+6}{16}=\tfrac{11}{16}$. It then catches it half the time, so the catch probability is $\tfrac{11}{32}$ and the complement is $\tfrac{21}{32}$.