Player A needs to flip Heads twice to win (does not need to be consecutive Heads), while Player B only needs to flip Heads once to win. Player A goes first. They take turns flipping coins until one player wins. How many games on average would player A win after 100 games? Each win counts as one game, regardless of how many rounds the game lasts.
Here are some examples:
(Round 1) Player A flips Heads; Player B flips Tails.
(Round 2) Player A flips Heads (Player A wins)
(Round 1) Player A flips Heads; Player B flips Heads (Player B wins)
(Round 1) Player A flips Tails; Player B flips Heads (Player B wins)
(Round 1) Player A flips Heads; Player B flips Tails
(Round 2) Player A flips Tails; Player B flips Tails
(Round 3) Player A flips Heads (Player A wins)
The purpose of the problem is to determine your likelihood of winning in a real game-theory situation. There are many sports example and one of them is pick-up basketball. For example, the first team that scores 21 points wins. Your team has 17 points while the other team has 19. Assuming both sides have a 50% chance of scoring a basketball when any player shoots, how often would the team with 17 points win? The coin flip analogy is a simplified version of this problem.
I've tried doing simulations (below) but don't think that is the right way to solve the problem.
HT HTH HTTH