Player A has one more coin than player B. Both players throw all of their coins simultaneously and observe the number that come up heads. Assuming all the coins are fair, what is the probability that A obtains more heads than B?
Either A throws more heads than B, or A throws more tails than B, but (since A has only one extra coin) not both.
By symmetry, these two mutually exclusive possibilities occur with equal probability.
Therefore the probability that A obtains more heads than B is ½. It is perhaps surprising that this probability is independent of the number of coins held by the players.
If player A has m coins and player B has n, what is the probability that A will throw more heads than B?
Source: The Theory of Gambling and Statistical Logic, by Richard A. Epstein. See Chapter 4.