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等分并不更常见：恰好 n:n 的概率

Head and Tail

专题
Probability / 概率
难度
L4

题目详情

According to many people's intuition, when two events, such as head and tail in coin tossing, are equally likely then the probability that these events will occur equally

often increases with the number of trials. This expectation reflects the intuitive notion that in the long run, asymmetries of the frequencies of head and tail will "balance out" and cancel.

To find the basis of this intuition, consider that 2n2n fair and independent coins are thrown at a time.

a. What is the probability of an even nn : nn split for head and tail when 2n=202n = 20 ?

b. Consider the same question for 2n=2002n = 200 and 2n=20002n = 2000 .

解析

抛 2n2n 枚公平硬币，恰好 nn 正面的概率为

p(2n)=(2nn)2−2n.p(2n)=\binom{2n}{n}2^{-2n}.

(a) 当 2n=202n=20（n=10n=10）时

p(20)=(2010)2−20≈0.1762.p(20)=\binom{20}{10}2^{-20}\approx 0.1762.

(b) 当 nn 很大时，用 Stirling 近似

(2nn)2−2n∼1πn.\binom{2n}{n}2^{-2n}\sim \frac{1}{\sqrt{\pi n}}.

因此

  • 2n=2002n=200（n=100n=100）：p(200)≈1100π≈0.0564p(200)\approx \frac{1}{\sqrt{100\pi}}\approx 0.0564
  • 2n=20002n=2000（n=1000n=1000）：p(2000)≈11000π≈0.0178p(2000)\approx \frac{1}{\sqrt{1000\pi}}\approx 0.0178

结论：随着试验次数增大，“恰好对半”反而变得更不可能。