HTH before HHT

HardProbability Random Variables~9m

You and a friend flip a fair coin repeatedly, recording the sequence. You win if the pattern $HTH$ appears before $HHT$ ; otherwise your friend (who has $HHT$ ) wins. What is the probability that your friend wins?

Show hints (2)+

Set up a Markov chain on the recent flips ( $H$ , $HH$ , $HT$ ). Note that reaching $HH$ guarantees $HHT$ eventually.
Solve $f=\tfrac12+\tfrac12 g$ , $g=\tfrac12 f$ . Pattern races aren't symmetric!

Answer

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0.6667 (± 0.001)

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Asked at: Jane Street, Citadel

HTH before HHT

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