Hard

Probability

You're practicing soccer by taking $100$ penalty kicks. Assume that you have made the first goal but missed the second. For each of the following kicks, the probability that you score is the fraction of goals you've made thus far. For example, if you made 17 goals out of the first 30 attempts, then the probability that you make the 31st goal is $\frac{17}{30}$. After 100 attempts, including the first two, what is the probability that you score exactly $66$ penalty kicks?

Notes

Hint