Flips of a biased coin until the first head

Question

A biased coin lands heads with probability $0.3$ on each flip, independently. You flip it over and over until the first head appears.

On average, how many flips does that take (counting the flip that lands heads)?

Accepted Answer

Each flip is an independent trial with success probability p = 0.3. We want the expected number of flips until the first head. The first flip costs 1. With probability p it is heads and we stop; with probability 1-p it is tails and we start over, needing an expected E more flips: E = 1 + (1-p)\,E \quad\Longrightarrow\quad E = \frac{1}{p}. With p = 0.3, E = \frac{1}{0.3} = \frac{10}{3} \approx 3.33 	ext{ flips.} Takeaway. Expected flips to the first head is 1/p = 1/0.3 = 10/3 \approx 3.33.

Flips of a biased coin until the first head

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