I recall reading a puzzle that I believe was about shooting a basketball, where the probability of success for a given trial changes depending on previous successes. I searched around for it but had trouble finding it. I'm more interested in the name of the puzzle and some references than in the analysis of the problem itself.
It goes something like this:
You're shooting free throws in basketball, and the probability of success depends on how many free throws you've made so far. In particular, on the $i$-th shot, you have a $\frac{1+s}{1+i}$ of making the basket, where $s$ is the number of previous successes.
Given that you take $n$ shots, what is the probability of making exactly $k$ of them?
Example
There is a $\frac12$ chance of making (or missing) the first shot.
If you make the first shot, then there is a $\frac23$ chance of making the second shot.
If you make the first two shots, then there is a $\frac34$ chance of making the third shot.
However, if you make the first shot and miss the second shot, then there is a $\frac24$ chance of making the third.
