Subsequence of coin tosses

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?

ronret45 Expert Asked on 1st August 2015 in Microsoft Interview Puzzles.
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