Average flips to see the first head

Question

You flip a fair coin repeatedly, stopping the moment it lands heads. What is the expected number of flips, counting the flip that lands heads?

Accepted Answer

Let E be the expected number of flips. The first flip always occurs (that is the "1"). With probability 	frac12 it is heads and you stop; with probability 	frac12 it is tails and you start over, needing E more on average: E = 1 + 	frac12 E \quad\Longrightarrow\quad 	frac12 E = 1 \quad\Longrightarrow\quad E = 2. Takeaway. A fair coin lands heads with probability 	frac12, so the mean wait is 1/p = 2 flips.

Average flips to see the first head

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