Flips to First Head

Question

On average, how many times must you flip a fair coin to see the first head?

Accepted Answer

Let E be the expected number of flips. Condition on the first flip: with probability frac12 it's a head (done in 1 flip); with probability frac12 it's a tail and you've spent one flip and are back where you started (memorylessness): E = frac12(1) + frac12(1 + E) implies frac12 E = 1 implies E = 2. In general a geometric random variable with success probability p has mean frac1p, here frac11/2 = 2. The conditioning trick ("first-step analysis") is the master key for this entire family of questions; the harder versions (first HH, first 6 then 6 again, etc.) all start exactly this way.

Flips to First Head

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