Answer to puzzle 13: Coin triplets
a. What is the optimal strategy for each player? With best play, who is most likely to win?
The second player can always choose a triplet that has a better than 1/2 probability of appearing before the first player's triplet.
The best strategy for player 1 is to limit player 2's probability of winning to 2/3. This can be done by choosing triplet HTH or HTT, or their mirror images, THT or THH.
Player 2 should choose a triplet that maximizes the chance of winning, based upon player 1's choice. In each case, this is a unique choice, as follows:
1 plays HHH, 2 plays THH and wins with probability 7/8;
1 plays HHT, 2 plays THH and wins with probability 3/4;
1 plays HTH, 2 plays HHT and wins with probability 2/3;
1 plays HTT, 2 plays HHT and wins with probability 2/3.
b. Suppose the triplets were chosen in secret? What then would be the optimal strategy?
An optimal strategy for both players is to choose at random, with probability 1/2 for each, between HTT, and its mirror image, THH.
c. What would be the optimal strategy against a randomly selected triplet?
We must decide what to do if our play matches the randomly selected triplet. We may call this void and play again, or we may split the (notional) winnings. The decision does not affect our choice of best play, but it does slightly alter the expected return from each play.
The best play against a random triplet is HTT, or its mirror image, THH. The table below shows the expected return and percentage expected profit from each play.
| Play | Expected return (void) | % expected profit (void) | Expected return (split) | % expected profit (split) |
|---|---|---|---|---|
| HHH | 317/840 | −12.3 | 377/960 | −10.7 |
| HHT | 469/840 | 5.8 | 529/960 | 5.1 |
| HTH | 407/840 | −1.5 | 467/960 | −1.4 |
| HTT | 487/840 | 8.0 | 547/960 | 7.0 |