First to three of a kind in a row

MediumProbability Expected Value~6m

Anna wins the moment three heads appear in a row; Brenda wins the moment three tails appear in a row. You flip a fair coin until one of them wins. What is the expected number of flips?

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By symmetry track just the current run length $k$ ; a mismatch resets you to run length $1$ , not $0$ .
Solve $e_1=1+\tfrac12 e_2+\tfrac12 e_1,\ e_2=1+\tfrac12 e_1$ to get $e_1=6$ , then add the first flip. (Or $2^3-1$ .)

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Asked at: Jane Street, SIG

First to three of a kind in a row

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