Expected number of coin flips until the first head

Question

You flip a fair coin repeatedly until it lands heads for the first time. What is the expected number of flips? Follow-up: what if the coin is biased and lands heads with probability p?

Accepted Answer

Let E be the expected number of flips until the first head. Condition on the first flip: - With probability 	frac{1}{2} it's a head, and we're done after 1 flip. - With probability 	frac{1}{2} it's a tail: we've used 1 flip and, because the flips are independent, we're back exactly where we started, expecting E more flips. This gives the equation E = 	frac{1}{2}(1) + 	frac{1}{2}(1 + E) Solving: E = 1 + 	frac{1}{2}E, so 	frac{1}{2}E = 1, which means E = 2. Biased coin. The same conditioning with head-probability p gives E = p(1) + (1-p)(1+E) \implies E = \frac{1}{p}. This is the mean of …

Expected number of coin flips until the first head

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