Basketball scores

A basketball player practices free throws. He takes exactly $100$ shots, scoring one point for each basket and zero for each miss. He makes his first shot and misses his second. After that, the probability of making each shot equals the fraction of his previous shots that he has made. For example, if he has made $23$ of his first $40$ shots, the probability that he makes his $41^{st}$ shot is $\frac{23}{40}$. What is the probability that he makes exactly $50$ baskets out of all $100$ shots (including the first two)?