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Alice Loses a Friend


Alice and Bob are flipping two coins randomly, with Alice going first with both flips. If Bob matches both of Alice's coins, then Bob wins two coins from Alice. If Bob does not match both of Alice's coins, then Alice wins two coins from Bob.

Note: In this game, order does not matter (i.e. HT = TH)...just the number of Heads and Tails.

Question 1: Is this game fair? Also, what is Alice's expected value per game?

Question 2: NOW, repeat the game where Alice and Bob flip three coins with same rules. Is this game fair? Also, what is Alice's expected value per game?

Question n: Generalize the game, so that Alice and Bob are flipping n coins each. Is this game fair? Also, what is Alice's expected value per game?

 

Source: From J. Kurtzke's "Tossing Coins: Who Should Go First?" Oregon Math Teacher, Oct. 1995


Hint: Set up a tree diagram. From first node, three branches should represent Alice's flips (HH, HT, TT). From each of these nodes, two branches should represent Bob's results (match or no match). Then, by multiplying and summing the probabilities, the expected value can be found.

 


Solution Commentary: For the first game, Alice's expected value is 2[3/16 + 1/4 + 3/16] - 2[1/16 + 1/4 + 1/16] = 1/2.

But, for the second game... or the nth game....