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# Coin Triplets

## Question

Two players play the following game with a fair coin. Player 1 chooses (and announces) a triplet (HHH, HHT, HTH, HTT, THH, THT, TTH, or TTT) that might result from three successive tosses of the coin. Player 2 then chooses a different triplet. The players toss the coin until one of the two named triplets appears. The triplets may appear in any three consecutive tosses: (1st, 2nd, 3rd), (2nd, 3rd, 4th), and so on. The winner is the player whose triplet appears first.
What is the optimal strategy for each player? With best play, who is most likely to win?
Suppose the triplets were chosen in secret? What then would be the optimal strategy?
What would be the optimal strategy against a randomly selected triplet?
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Player 1 picks a triplet T1.

Player 2 picks a triplet T2.

Coin is tossed until T1 or T2 appears and wins.

Every triplet T1 has an evil twin T2 that will beat it on average.

Suppose T1 = TTT.

If TTT is not the result of the first three flips, it will be preceded by H and lose to HTT.

TTT is the result of the first three flips 1/8 of the time.

Thus TTT loses to HTT with 7:1 odds.

Other pairwise triplet probabilities can be deduced similarly.
The table gives P2's winning probability for every triplet combination T1, T2.
The bottom two rows give the best and average winning probability for each T1

What is the optimal strategy for each player?

P1 should pick T1= HTH, HTT, THH or THT.
The best that P2 can achieve then is a winning probability of .667
,

That is the smallest best winning probability among the possible T1 choices.

P2 should look in the column headed by T1 and choose the T2 that has the highest winning probability.

With best play, who is most likely to win?

P2 can always gain favorable odds, ranging from 2:1 to 7:1 depending on T1

Suppose the triplets were chosen in secret?

I interpret this to mean P2 chooses T2 without knowing T1.

Nothing different for P1. He should still choose
HTH, HTT, THH or THT.

P2 does not know T1, so he should seek the T2 with best average winning probability.
These numbers are given in the last column of the table.

The choices are either HTT or THH with average winning probability of .507.

HTT loses only to HHT; draws against itself, HTH, THH and THT; wins against HHH, TTH and TTT.
THH loses only to TTH; draws against itself, THT, HTT and HTH; wins against TTT, HHT and HHH.

What would be the optimal strategy against a randomly selected triplet?

I take this to ask P1's optimal strategy in the case that T2 will be chosen at random.

P1 should look at the bottom row of the table to see P2's winning probability averaged over all the T2s.

P1 should thus choose HTT or THH.

This gives T2 the lowest average winning probability: .368.

P2\P1| HHH HHT HTH HTT THH THT TTH TTT | MAX | AVG |
---\-+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
HHH | --- .500 .400 .400 .125 .417 .300 .500 |.500 |.330 |
HHT |.500 --- .667 .667 .250 .625 .500 .700 |.700 |.489 |
HTH |.600 .333 --- .500 .500 .500 .375 .583 |.600 |.424 |
HTT |.600 .333 .500 --- .500 .500 .750 .875 |.875 |.507 |
THH |.875 .750 .500 .500 --- .500 .333 .600 |.875 |.507 |
THT |.583 .375 .500 .500 .500 --- .333 .600 |.600 |.424 |
TTH |.700 .500 .625 .250 .667 .667 --- .500 |.700 |.489 |
TTT |.500 .300 .417 .125 .400 .400 .500 --- |.500 |.330 |
-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
MAX |.875 |.750 |.667 |.667 |.667 |.667 |.750 |.875 |
-----+-----+-----+-----+-----+-----+-----+-----+-----+
AVG |.545 |.386 |.451 |.368 |.368 |.451 |.386 |.545 |
-----+-----+-----+-----+-----+-----+-----+-----+-----+

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