fun probability question from an old Putnam

[gumpzilla]

Saw this question yesterday, and I liked it quite a bit:

A basketball player sets out to shoot 100 free throws. He makes his first, and misses his second. From that point on, the probability that he makes free throw number N is equal to the percentage of his free throws that he's made already. So, on shot 3, he's 50/50 to hit. If he makes shot 3, then on shot 4 he will make his shot 2/3 of the time, etc.

After he's shot his 100 free throws, what is the probability that he made exactly 50?