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Find the Winning Strategy for a Coin Flip Game

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Your opponent specifies 3 successives results of tosses of a coin, e.g. HHT. You then specify another such result, e.g. THT. The winner is the person whose sequence appears first when a fair coin is tossed successively and independently. Find the strategy which will allow you, the second player, to win at least 2/3 of the time. More specifically, for each possible choice of your opponent, find the choice that gives you the maximum probability of winning and calculate that probability.

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Solution:
Let us consider a few cases for the successive and independent tosses. Assume that you opponent choose a sequence HHT.
First Case: If your opponent wins at the 3rd successive toss, then it is obvious that there is only one way to win, which is by having a pattern of HHT.
Second Case: If your opponent wins at the 4th successive toss, then there will be two possible patterns of HHHT and THHT.
Third Case: If your opponent wins at the 5th successive toss, then there will be four possible patterns of HHHHT, HTHHT, THHHT, and TTHHT.
Fourth Case: If your opponent wins at the 6th successive toss, then there will be seven possible patterns of HHHHHT, HTHHHT, HTTHHT, THHHHT, THTHHT, TTHHHT, and TTTHHT.
Consider closely to second, third, and fourth cases, if you chose a sequence THH, then you will reduce the winning chance of your opponent. More specifically, let begin with the ...

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