Even count until the coin repeats

EasyProbability~4m

You flip a fair coin until you first see two consecutive flips land the same (two heads in a row or two tails in a row). What is the probability that the total number of flips is even?

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Write $N=1+G$ where $G\sim\text{Geom}(\tfrac12)$ counts flips from the second on. $N$ even $\iff$ $G$ odd.
Sum $\tfrac12+\tfrac1{2^3}+\tfrac1{2^5}+\cdots=\tfrac{1/2}{1-1/4}=\tfrac23$ .

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0.6667 (± 0.005)

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Even count until the coin repeats

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