用偏硬币模拟公平硬币

Simulating a Fair Coin with an Unfair One

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

一枚硬币正面概率 $p=\frac{1}{7}$ ，反面概率 $q=\frac{6}{7}$ 。只用这枚硬币，设计方法模拟一次公平抛硬币（正反各 $1/2$ ）。

A coin is biased such that it comes up heads with probability $p = \frac{1}{7}$ and tails with probability $q = \frac{6}{7}$ . Using only this coin, design a method to simulate a fair coin, i.e., to get heads and tails with equal probability of $\frac{1}{2}$ .

解析

冯·诺依曼方法：连续抛两次。

若结果为 HT，则输出“正”。
若结果为 TH，则输出“反”。
若为 HH 或 TT，则丢弃并重来。

因为 $P(HT)=pq$ 与 $P(TH)=qp$ 相等，所以在条件事件 {HT,TH} 下输出正反各为 $1/2$ 。

Original Explanation

To simulate a fair coin using the biased coin, follow this procedure:

Flip the coin twice and observe the outcome.

There are four possible outcomes:

Now examine their probabilities:

$P(\text{HT}) = p \cdot q$
$P(\text{TH}) = q \cdot p$

Both HT and TH have the same probability, namely $pq$ .

So, define your fair coin outcome as follows:

If you get HT, declare the result as "Heads"
If you get TH, declare the result as "Tails"
If you get HH or TT, discard the outcome and flip again.

This method ensures that the resulting outcomes are equiprobable because $P(\text{HT}) = P(\text{TH})$ , thereby simulating a fair coin with a biased one.