A fair die is rolled indefinitely. Player A wins if two consecutive rolls show 3 or 5, and player B wins if two consecutive rolls show 1 or 2 or 4 or 6. The probability that player A wins in the long run is

Step-by-step solution

Let S be the event of rolling a 3 or 5 . The probability of this event is P(S) = 2 6 = 1 3 . Let F be the event of rolling a 1, 2, 4, or 6 . The probability of this event is P(F) = 4 6 = 2 3 . Let P be the probability that player A wins the game. Let P_S be the probability that player A wins given that the previous roll was S . Let P_F be the probability that player A wins given that the previous roll was F . From the start of the game, the first roll is either S or F : P = 1 3 P_S + 2 3 P_F If the previous roll wa