Wasted all morning solving this. I know the answer, but I’d love a really satisfying and intuitively understandable explanation of why the answer is the answer.
There’s a basketball player practicing free throws. He has the odd characteristic that once he’s made at least one free throw and missed at least one free throw, his probability of making any further free throw is precisely equal to the fraction that he has made so far this session. So if he’s taken 8 shots and has made 5 of them, he’s 5/8 likely to make the next one, 3/8 likely to miss it.
His coach watches him, and sees him make the first, and miss the second. The coach then leaves for a while, and comes back. When he comes back, the player has taken 98 throws. The coach then watches the player take throw #99, and make it. From the coach’s perspective, what is the probability that the player will make throw #100?