[I got this problem from Joe Morris, who created it.]

Each of two players
picks a different sequence of two coin tosses. That is, each player gets
to pick among HH, HT, TH, and TT. Then, a coin is flipped repeatedly and
the first player to see his sequence appear wins. For example, if one
player picks HH, the other picks TT, and the coin produces a sequence that
starts H, T, H, T, T, then the player who picked TT wins. The coin is
biased, with H having a ^{2}⁄_{3} probability and T having a ^{1}⁄_{3} probability. If
you played this game, would you want to pick your sequence first or second?

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