Problem 2 (Total 30 marks) Homer and Marge play a game by taking turns rolling a standard six-sided die, starting with Homer, until there occurs a

Problem 2 (Total 30 marks)

Homer and Marge play a game by taking turns rolling a standard six-sided die, starting with Homer, until there occurs a sequence of k or more 4s immediately followed by a 5 or 6 (e.g. 45, 245 or 5446, if k = 1). The last person to roll wins the game. (E.g., if k = 2, then each of the sequences 446, 64445 and 465443445 results in Homer winning.) Find the probability that Homer wins, for each k = 0, 1, 2, 3 and 99.