Free throws where accuracy follows past success

Frank's opening free throw goes in and his next one does not. From the third attempt on, the chance he makes any given shot equals the fraction of his earlier attempts that went in. What is the probability that he has made exactly $50$ of his first $100$ attempts?

$1/99$. This is a Pólya urn started with one make and one miss. For such an urn the number of additional makes in the next $m$ shots is uniform on $\{0,\dots,m\}$, so over the remaining $98$ shots each of the $99$ totals is equally likely, including exactly $49$ more makes.