A fair decision from a biased coin

Question

You need to settle a decision with a perfectly fair fifty-fifty chance, but the only coin available is biased: it lands heads with some probability p that is not one half, and you do not even know what p is (only that it is strictly between 0 and 1). 1. How do you use this coin to make a fair decision? 2. On average, how many flips does your method use, for example if p = 0.9?

Accepted Answer

Flip the coin twice. If it shows heads then tails, choose option A. If it shows tails then heads, choose option B. If it shows two heads or two tails, ignore the result and flip twice again. Why it is fair. Flips are independent, so the probability of heads-then-tails is p(1 minus p), and the probability of tails-then-heads is (1 minus p)p. These are equal, whatever p is. Every round either ends with one of these two equally likely outcomes or is discarded, so conditional on ending, the two options are exactly 50/50. This trick is due to John von Neumann. How long it takes. Each pair of flips…

A fair decision from a biased coin

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