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Even sums only

Expected valueMedium

You pay $4$ dollars to roll $2$ fair $6$ -sided dice. You receive the sum in dollars if the sum is even, and nothing if it is odd. What is your expected net result?

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Hint

List the even sums with their numbers of ways.

Solution

The even sums $2$ , $4$ , $6$ , $8$ , $10$ , $12$ occur in $1$ , $3$ , $5$ , $5$ , $3$ , $1$ ways out of $36$ .
The expected payout is $\frac{2 + 12 + 30 + 40 + 30 + 12}{36} = \frac{126}{36} = \frac{7}{2}$ .
The net result is $\frac{7}{2} - 4 = - \frac{1}{2}$ .

Answer: $- \frac{1}{2}$

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