You play against your friend in a coin flipping game, where the objective is to get the most heads after three coin flips. A player wins if they have more heads than the opponent. If the numbers of heads are equal, then no one wins; it is a tie. You will take turns flipping coins, and your friend flips first. You want to cheat, so you created a fake coin that looks identical to a real, fair coin. The fake coin has a 80% chance of getting heads. Your friend is suspicious, so your friend gets to pick first from the two coins randomly, with equal probability. You are forced to take the other coin. You will both use your own coin for all three coin flips. The game begins, and your friend flips the coin once and gets heads.
(a)Given that your friend’s first coin flip returns heads, what is the probability that your friend got the fake coin, to your disadvantage?
(b)Given that your friend’s first coin flip returns heads, what is the probability that you will win the game?
I don't know what did I do wrong in part b as well as how to use part a's answer.
Me get real coin: Me get 2 head, friend gets 1 or Me get 3 head, friend gets 2;+
Me get fake coin: Me get 2 head, friend gets 1 or Me get 3 head, friend gets 2.