Shooting where success rate equals past hit fraction

A basketball player takes $100$ shots. The first goes in and the second misses. From then on, the chance of making each shot equals the proportion of shots made so far. What is the probability that the player finishes with exactly $50$ made shots?

$1/99$. This is a Pólya urn: by induction, after $n$ shots the number made is uniform on $\{1,\dots,n-1\}$. It holds for $n=2$, and if made $=k$ with probability $\tfrac1{n-1}$ for each $k$, the next shot preserves uniformity on $\{1,\dots,n\}$. So after $100$ shots each of $1$ to $99$ has probability $\tfrac1{99}$.