Free throws where success rate copies past accuracy

A player makes his first free throw and misses his second. From the third attempt on, the chance of making each shot equals the fraction of his previous attempts that he made. How many of his first $100$ attempts does he make, on average?

$50$. This is a Pólya urn starting with one make and one miss. The expected success fraction is a martingale: if $E_{n-1}=\tfrac{n-1}2$, then $E_n=E_{n-1}+\tfrac{E_{n-1}}{n-1}=\tfrac n2$. So after 100 attempts he averages 50 makes.