投篮命中率随历史命中率变化：恰好 50 中的概率

Basketball Scores

专题: Strategy / 策略
难度: L4
来源: QuantQuestion

题目详情

一名篮球运动员投 100 次罚球。她第 1 次投中（当前命中数 1），第 2 次投失（命中数仍为 1）。

之后对第 $n$ 次投篮（ $n\ge 3$ ），她投中的概率等于“截至目前的命中比例”（命中数 / 已投次数）。

投完 100 次后，问：她最终恰好命中 50 球的概率是多少？

A basketball player takes 100 free throws. She makes the first one (score = 1), misses the second one (still total = 1). For each subsequent throw $n$ , the probability of scoring is the fraction of total successes so far. If after 100 throws, we look at the total made baskets, what is the probability that she ends up with exactly 50 made baskets?

解析

这是一个 Pólya’s urn 过程。可证明对 $2\le n\le 100$ ，命中数 $k$ 在 $1\le k\le n-1$ 上满足

P_{n,k}=\frac{1}{n-1}.

因此

P_{100,50}=\frac{1}{99}.

Original Explanation

Let $P_{n,k}$ = probability of having $k$ successes after $n$ throws. One can show (by a combinatorial/inductive argument) that for $2 \le n \le 100$ and $1 \le k \le n-1$ , $P_{n,k} = \frac{1}{n-1}.$ Hence, specifically, $P_{100,50} = \frac{1}{99}.$