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It belongs to the family of non-transitive games in which choosing second is not a disadvantage, but rather a way of responding to the rival sequence with a better one.
Two players take turns choosing a sequence of three coin outcomes. A fair coin is then tossed repeatedly until one of the two sequences appears for the first time as a consecutive block.
Whoever chose that sequence wins. The first player has chosen:
face, face, face You choose later.
What sequence should you choose to maximize your chances of winning?
The optimal response to heads-heads-heads is to choose tails-heads-heads. Because?
Because the first player only wins if the game starts directly with heads-heads-heads on the first three tosses. That happens with probability 1/8.
On the other hand, if a cross appears before three consecutive heads come up at the beginning, the sequence cross-head-head is very well positioned to appear before heads-head-heads. In fact, the choice tails-heads-heads wins with probability 7/8.
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