Two dice, win the sum plus a bonus for doubles

Question

You roll two fair six-sided dice. You win the sum of the two dice in dollars, and if the two dice show the same number (doubles) you win an extra 6 bonus. On average, how much do you win per roll?

Accepted Answer

The payout has two pieces: the sum, and a bonus that only sometimes applies. Handle them separately and add. The sum. Each die averages 3.5 (the middle of 1 through 6). The average of a total is the total of the averages, so two dice average: 3.5 + 3.5 = 7. The doubles bonus. Doubles means both dice match. There are 6 matching outcomes (1-1, 2-2, up to 6-6) out of 6 	imes 6 = 36 equally likely rolls, so the chance of doubles is 6/36 = 1/6. The bonus is 6, but you only collect it one-sixth of the time, so on average it is worth: 6 	imes 	frac{1}{6} = 1. Add the two pieces: EV = 7 + 1 = 8. T…

Two dice, win the sum plus a bonus for doubles

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