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Capped flipping

Expected valueEasy

A fair coin is flipped until the first head appears, but you stop after at most $3$ flips. What is the expected number of flips?

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Hint

Flip number k happens only if all earlier flips were tails.

Solution

The first flip always happens, so it adds $1$ .
The second flip happens only if the first was a tail, with probability $\frac{1}{2}$ .
The third flip happens only if the first $2$ were tails, with probability $\frac{1}{4}$ .
By linearity the expected number of flips is $1 + \frac{1}{2} + \frac{1}{4} = \frac{7}{4}$ .
The answer is $\frac{7}{4}$ .

Answer: $\frac{7}{4}$