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Paying per flip

Expected valueEasy

A fair coin is flipped repeatedly until the first head appears. Each flip costs $3$ . What is the expected total cost?

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Hint

First find the expected number of flips until the first head.

Solution

The number of flips until the first head is geometric with success probability $\frac{1}{2}$ .
Its expected value is $1 \div (\frac{1}{2}) = 2$ flips.
Each flip costs $3$ , so by linearity the expected cost is $2 \times 3$ .
The expected total cost is $6$ .

Answer: $6$