A little variation to the Biased coin puzzle.
You and your rival are competing for the same girl, and decide to settle it with a coin toss. Your rival has known the girl longer than you have, so you agree that it is fair for him to have a chance of winning equal to P, where P > 0.5. However, you only have a fair coin.How can you conduct this contest such that the biased probability is manifested? What is the average number of coin flips needed to determine a winner?