Answer Happy

Please answer the question with explanations.
(a) Amir and Brigitte play a card game. Amir starts with a hand of 6 cards: 2 red, 2 yellow and 2 green. Brigitte starts with a hand of 4 cards: 2 purple and 2 white. Amir plays first. Amir and Brigitte alternate turns. On each turn, the current player chooses one of their own cards at random and places it on the table. The cards remain on the table for the rest of the game. A player wins and the game ends when they have placed two cards of the same colour on the table. Determine the probability that Amir wins the game. (b) Carlos has 14 coins, numbered 1 to 14. Each coin has exactly one face called "heads”. When flipped, coins 1,2,3,..., 13, 14 land heads with probabilities hi, h2, h3,..., h13, h14, respectively. When Carlos flips each of the 14 coins exactly once, the probability that an even number of coins land heads is exactly Ž. Must there be a k between 1 and 14, inclusive, for which hk = }? Prove your answer. (c) Serge and Lis each have a machine that prints a digit from 1 to 6. Serge's machine prints the digits 1, 2, 3, 4, 5, 6 with probability P1, P2, P3, P4, P5, P6, respectively. Lis's machine prints the digits 1, 2, 3, 4, 5, 6 with probability 91, 92, 93, 94, 95, 96, respectively. Each of the machines prints one digit. Let S (i) be the probability that the sum of the two digits printed is i. If S(2) = S(12) = {$(7) and S(7) > 0, prove that P1 = Pc and qı = 46-