Coin Flips

In the ordered flip sequence HTTHHTHHHTHTTHHT, the four-flip sequence THHT occurs before HTHT. Two people could make a bet, each choosing a four-flip sequence, and then flip a coin until one of the sequences occurs. In the game, which sequence is best? Consider the two-flip sequences HH and TH. After three flips, the possibilities are HHH HHT HTH HTT THH THT TTH TTT. In these, HH appears first twice, while TH appears first four times. Already, TH has the advantage. This Demonstration shows how the four-flip sequences compare, with TTTT and HHHH left out, since neither is ever a good bet. Notice the counterclockwise ring of arrows on the outside—every flip sequence can win against at least one other. Thus, if you allow your victim to pick any sequence of four, you can always pick something better. The darkness of an arrow indicates how often one flip result will win against another.