Three players, A, B, and C, each put ten dollars into a pot with a list on which they have, independently of one another, predicted the outcome of three successive tosses of a fair coin. The fair coin is then tossed three times. The player having most correctly predicted the three outcomes gets the contents of the pot. The contents are to be divided if multiple players guess the same number of correct outcomes.
(a) Calculate the expected value of the amount that player A will get.
(b) Suppose that players A and B decide to collaborate, unbeknownst to player C. The collaboration consists of the two players agreeing that the list of player B will always be a mirror image of player A's list (should player A predict an outcome of HTT , for example, then player B would predict TTH). Calculate the expected value of the amount that player A will receive.