After finding a coin in her farm, Alice proposes the following game to Bob:
Alice will flip the coin $n$ times. Bob will then flip the coin $2n$ times. If Bob manages to get the same amount of heads as Alice, he wins. Otherwise Alice wins.
As they're best friends, Alice wants the game to be completely fair. Assuming Bob pays a dollar to play, how much should she pay Bob for a win so they're expected to break even?