Stathis Papaioannou writes:
> In the new casino game Flip-Flop, an odd number of players pays $1 each to
> individually flip a coin, so that no player can see what another player is
> doing. The game organisers then tally up the results, and the result in the
> minority is called the Winning Flip, while the majority result is called the
> Losing Flip. Before the Winning Flip is announced, each player has the
> opportunity to either keep their initial result, or to Switch; this is then
> called the player's Final Flip. When the Winning Flip is announced, players
> whose Final Flip corresponds with this are paid $2 by the casino, while the
> rest are paid nothing.

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Think about if the odd number of players was exactly one. You're guaranteed
to have the Winning Flip before you switch.
Then think about what would happen if the odd number of players was three.
Then you have a 3/4 chance of having the Winning Flip before you switch.
Only if the other two players' flips both disagree with yours will you not
have the Winnning Flip, and there is only a 1/4 chance of that happening.
Hal Finney