Subsequence of coin tosses

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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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    he probability of the sequence HH is (2/3)*(2/3) = 4/9
    The probability of the sequence HT is (2/3)*(1/3) = 2/9
    The probability of the sequence TH is (1/3)*(2/3) = 2/9
    The probability of the sequence TT is (1/3)*(1/3) = 1/9.
    So if we look on this carefully we can see that if on the first toss we get H than if we choose a combination that starts with H we have 50% chance of winning (HH is 4/9 and HT is 2/9),
    and if the first toss is F and we choose a combination that starts with F we also have a 50% chance of winning (FH is 2/9 and FF is 1/9) so it seems like it doesn’t matter who picks first, but since the probability of HH is 44% and double or more that the other choices I do want to pick the sequence first.

    Moshe Expert Answered on 27th March 2023.
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