1) Alice and Bob are playing a game where they each write their name on a card. Alice goes first and chooses one of the two cards with equal probability. If Alice chooses her own card, then she wins. If she chooses Bob's card, then it's Bob's turn to play. He chooses one of the two cards randomly with equal probability, and he wins if gets his own card. Otherwise, the turn passes back to Alice and she tries again. This pattern continues until one or the other has win. Calculate the probability that Alice will win this game.
2) Claire walks up and wants to join the game so she adds a card with her name into the pile.Now, if Alice draws Bob's card then he goes next and if she draws Claire's card then Claire goes next. At each turn, the person drawing a card will win if they draw their own card, and otherwise the turn will pass to whoever is named on the card. Every card is drawn independently with all three cards having equal probability. They continue to take turns until somebody wins. Calculate the probability of Alice winning, the probability of Bob winning, and the probability of Claire winning.