Biased to Unbiased

If you have an unfair coin, which may be biased toward either heads or tails at an unknown probability, can you generate even odds using this coin?

Answer

Yes. The method is as follows:

  1. Toss the unfair coin twice.

  2. If the outcome is Heads-Tails (HT), declare "player 1 wins".

  3. If the outcome is Tails-Heads (TH), declare "player 2 wins".

  4. If the outcome is Heads-Heads (HH) or Tails-Tails (TT), ignore this pair of tosses and repeat the process.

This procedure yields even odds because:

\[ P(\text{HT}) = p(1-p) \quad \text{and} \quad P(\text{TH}) = (1-p)p, \]

which are equal regardless of the unknown bias \(p\). Outcomes HH and TT are irrelevant as they are discarded, ensuring unbiased results for HT vs. TH.

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