Problem

A coin has unknown head probability \(p\). Give a \(\text{two}\text{-}\text{flip}\) von Neumann rule for an unbiased \(A/B\) result, identify rerolls, and prove the accepted outcomes have equal probability.

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Give HH to A and TT to B and reroll mixed pairs: accepted probabilities are p² and (1-p)², so final chances are generally unequal.

Use one flip, H for A and T for B: chances are p and 1-p with acceptance 1, so the method is fair only when p=1/2.

Give the more frequent outcome to one player and the other outcome to the second: chances remain p and 1-p, so the method is generally biased.

Flip twice: HT gives A, TH gives B, and HH or TT rerolls. HT and TH both have probability p(1-p), so final chances are equal.

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