Pepys-Newton：哪种彩票更可能

Isaac Newton Helps Samuel Pepys

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

Pepys wrote Newton to ask which of three events is more likely: that a person get (a) at least 1 six when 6 dice are rolled, (b) at least 2 sixes when 12 dice are rolled, or (c) at least 3 sixes when 18 dice are rolled. What is the answer?

解析

设 $X_n\sim\mathrm{Bin}(n,1/6)$ 为 $n$ 次掷骰得到 6 的个数。

三件事分别为：

(a) $P(X_6\ge 1)$
(b) $P(X_{12}\ge 2)$
(c) $P(X_{18}\ge 3)$

直接计算（或用补事件）可得数值大小关系：

P(X_6\ge 1)\approx 0.6651> P(X_{12}\ge 2)\approx 0.6187> P(X_{18}\ge 3)\approx 0.5973.

因此最可能的是 $\boxed{(a)}$ 。