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抛 100 次恰好 50 正面的概率

Even Split

专题
Probability / 概率
难度
L4

题目详情

When 100 coins are tossed, what is the probability that exactly 50 are heads?

解析

设 X∼Bin(100,1/2)X\sim\mathrm{Bin}(100,1/2)，所求为

P(X=50)=(10050)2−100.\boxed{\mathbb{P}(X=50)=\binom{100}{50}2^{-100}}.

用 Stirling 近似可得

P(X=50)≈150π≈0.0796.\mathbb{P}(X=50)\approx \frac{1}{\sqrt{50\pi}}\approx 0.0796.