100 块饼干分 N 颗巧克力豆：至少 90% 都有豆

chocolate chip cookies

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

You are making chocolate chip cookies. You add $N$ chips randomly to the cookie dough, and you randomly split the dough into 100 equal cookies. How many chips should go into the dough to give a probability of at least $90\%$ that every cookie has at least one chip?

解析

把每颗巧克力豆独立等概率落入 100 块饼干之一（经典占位/投球入箱模型）。

令 $E$ 为“存在至少一块饼干没有豆”。用并集界：

\mathbb{P}(E)\le 100\cdot\mathbb{P}(\text{指定某块无豆})=100\left(\frac{99}{100}\right)^N.

要使 $\mathbb{P}(\text{每块至少 1 颗})\ge 0.9$ ，只需令右边 $\le 0.1$ ：

100\left(\frac{99}{100}\right)^N\le 0.1 \Rightarrow \left(\frac{99}{100}\right)^N\le 10^{-3} \Rightarrow N\ge \frac{\ln 1000}{\ln(100/99)}\approx 687.4.

因此取

\boxed{N=688}

即可保证（该上界是充分条件；更精细的近似会给出接近 686–690 的阈值）。