9 In the game of Rock, Paper, Scissors, two players each choose one option out of rock, paper and scissors, such that rock beats scissors, scissors beat paper, and paper beats rock. If a player wins a round, then they receive 1 point, and their opponent receives 0 points. If the two players choose the same option, then they each receive 0 points. Assume both players pick their option uniformly at random. 9(a) The two players decide to play 5 rounds. If player 1 wins the first round, what is the probability that they win the best-of-5 game? Justify your answer. 9(b) The players instead decide to play a sudden death variant, where the first player to win a round, wins the game. What is the expected number of rounds? Justify your answer. 9(c) Player 2 changes tactic and decides to instead randomly play rock 40% of the time, paper 40% and scissors 20%. If player 1 knows player 2's new tactic, then how should player 1 change their own probabilities of choosing each of rock, paper and scissors? Show that this choice maximises the probability that player 1 wins, and minimises the probability that player 2 wins