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Die plus coin bonus

Expected valueEasy

You roll a fair $6$ -sided die and flip a fair coin independently. Your payout is the die result plus $6$ if the coin shows heads (nothing for tails). What is the variance of the payout?

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Hint

Variances of independent parts add.

Solution

The die has variance $\frac{35}{12}$ .
The bonus is $6$ times a fair indicator of variance $\frac{1}{4}$ , so its variance is $36 \times \frac{1}{4} = 9$ .
The parts are independent, so the variance is $\frac{35}{12} + 9 = \frac{143}{12}$ .

Answer: $\frac{143}{12}$

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