Suppose you have agreed to settle a dispute with cousin Joe by tossing a coin. The problem is that neither of you has any change. Joe suggests that you instead toss a bottle cap, which will count as heads if it lands with the top up, and tails otherwise. As you cannot assume that these are equally likely, is there any way in which fairness can be guaranteed?

-————– Solution :

You can suggest a trick invented by computer pioneer John von Neumann. Instead of tossing the cap once and observing heads or tails, the cap is tossed twice. If this gives the sequence HT, you win; if it gives TH, Joe wins. If it gives HH or TT, nobody wins and you start over.

Suppose the the probability of heads is some value p, not necessarily 1/2. As the probability of tails is then 1 - p, independence gives that the probability to get HT is p (1 - p ) and the probability to get TH is (1 - p ) p, the same. The procedure is fair (but may take a while if p is very close to 0 or 1).