Consider a two-player game between Alice and Bob, in which the players take turns to roll a fair six-faced die. Alice rolls the die first. Then Bob rolls the die and he wins if he gets the same outcome as Alice. Otherwise, Alice rolls the die again and she wins if she gets the same outcome as Bob. The game continues in this way, and it terminates as soon as one player gets the same outcome as obtained by the opponent in the previous roll of the die. The player who succeeds in doing so first is the winner.
- Find the probability that the game does not terminate after the first three rolls (two by Alice and one by Bob) of the die.
- What is the probability that Alice will win the game?