Penney's coin-sequence game

Question

You and a friend each name a sequence of three coin-flip outcomes. Then a fair coin is flipped over and over, and whoever's sequence shows up first (as three flips in a row) wins. Your friend names HHH. You get to name yours afterward, and you pick THH.

What is your probability of winning?

Accepted Answer

The key is to see when HHH can possibly win. HHH wins only if the very first three flips are H, H, H, which happens with probability \left(	frac12ight)^3 = 	frac18. Why only then? If the first three flips are not all heads, then a tail appears somewhere before HHH could ever complete. But look at what must happen just before any run of three heads: the flip immediately preceding that run is a tail (otherwise the run would have started earlier). That tail, followed by the two heads that begin the winning run, spells T, H, H, so your sequence THH completes first. In short, unless heads come…

Penney's coin-sequence game

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