Penalty streak with success rate equal to past rate

You will take $100$ penalty kicks. The first one goes in and the second one misses. From the third kick on, your chance of scoring equals the proportion of kicks you have scored so far (e.g. after $17$ goals in $30$ kicks, kick $31$ scores with probability $\tfrac{17}{30}$). What is the probability that your total number of goals after all $100$ kicks is exactly $66$?

$1/99$. This is a Pólya urn started with one success and one failure: every particular sequence with $k$ further goals in the remaining $98$ kicks has probability $\tfrac{k!\,(98-k)!}{99!}$, so the number of further goals is uniform on $\{0,\dots,98\}$. Scoring $66$ in total needs $65$ more, which has probability $\tfrac1{99}$.