Suppose in an independent game which has 2 players, player 1 and player 2, the probability of player 1 to win each game is $r$. To be the overall winner of the game, one of the players needs to win 2 more games than the other. What is the probability that player 1 will be the overall winner?
My sketch to solve the question: Note that to be the overall winner, one player should have won 2 games consecutively. So if player 1 is the winner, the outcome should either a draw, i.e. each player wins a game consecutively or player one won 2 games consecutively. But I am not sure how to start calculating.