A stubborn coin

Question

A biased coin lands heads with probability 0.2 on each flip. You flip it until it lands heads, then stop. On average, how many flips does it take to get your first head?

Accepted Answer

Each flip succeeds (heads) with probability p = 0.2, independently, so the number of flips until the first head is geometric with average one over the success probability: E[	ext{flips until first head}] = \frac{1}{0.2} = 5. The intuition matches: if heads shows up one time in five on average, then a fresh wait for heads takes about five flips. The rarer the success, the longer the wait, in exact inverse proportion. Takeaway. Expected tries to the first success is 	frac1p. Halving the success rate doubles the wait: at p = 0.2 you wait 5 flips on average.

A stubborn coin

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