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Biased coin game

Expected valueEasy

A biased coin shows heads with probability $\frac{1}{5}$ on each independent flip. You flip until the first head. Each flip costs $1$ , and when the first head appears you receive $10$ . What is your expected net profit?

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Hint

Net profit is the prize minus the expected cost of flipping.

Solution

The number of flips until the first head is geometric with success probability $\frac{1}{5}$ .
The expected number of flips is $1 \div (\frac{1}{5}) = 5$ .
The expected flipping cost is therefore $5$ .
The prize is always received, so the expected net profit is $10 - 5$ .
The answer is $5$ .

Answer: $5$

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