Problem 22. (3.0 %) Suppose you play a matching coin game with your friend
as follows. Both you and your friend have a coin. Fach time, you two reveal a
side (i.e. H or T) of your coin to each other simultaneously. If the sides match,
you WIN a 1 from your friend if sides do not match then you lose a 1 your
friend. You decide to go with the following strategy. Every time, you will toss
your unbiased coin independently of everything else, and you will reveal its
outcome to your friend (your friend does not know the outcome of your random
toss until you reveal it). Then,
(i) On average, you will lose money to your smart friend
(ii) On average, you will lose neither lose nor win. That is, your average
gain/loss is 0.
(iii) On average, you will make money from your friend
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