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Capped doubling prize

Expected valueMedium

You pay an entry fee of $4.5$ dollars. A fair coin is flipped up to $6$ times; if the first heads is on flip $k$ , you receive $2^{k}$ dollars, and if there is no heads in $6$ flips you receive nothing. What is your expected net result?

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Hint

Each possible first-heads flip contributes the same amount.

Solution

The first heads is on flip $k$ with probability $(\frac{1}{2})^{k}$ , contributing $(\frac{1}{2})^{k} \times 2^{k} = 1$ .
There are $6$ possible values of $k$ , so the expected payout is $6$ .
The net result is $6 - 4.5 = \frac{3}{2}$ .

Answer: $\frac{3}{2}$

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