+0 Alice and Bob are playing a game. Alice starts first. On Alice's turn, she flips a coin. If she gets a heads, she wins. If not, it becomes Bob's turn. On Bob's turn, he flips a coin. If he gets a tails, he wins. If not, it becomes Alice's turn. What is the probability that Alice wins the game?
+410 The probability that Alice wins a game is the probability she will win on her first turn + the probability that she will win on her second turn + the probability she will win on her turn turn, and so on...
Alice has a 12 probability of winning on their first turn, finishing the game.
Since whenever someone wins, the game is finished Bob has a 12⋅12=(12)2 probability of winning on their first turn.
Alice has a (12)2∗12=(12)3 probability of winning on her their second turn.
Bob has a (12)3∗12=(12)4 probability of winning on their second turn.
Alice has a (12)4∗12=(12)5 probability of winning on her thrid turn.
We see Alice has a (12)1+(12)3+(12)5+(12)7... chance of winning.
We can use the infinite geometric series fomula,121−14, we get 23.
