Two dice, win the sum plus ten dollars for a seven

Question

You roll two fair six-sided dice. You win the sum in dollars, and if the sum is exactly 7 you win an extra 10 bonus. On average, how much do you win per roll?

Accepted Answer

Split the payout into the sum and the bonus, average each, then add. The sum. Each die averages 3.5, so two dice average 3.5 + 3.5 = 7 dollars. The seven bonus. How often does the sum equal 7? Out of the 6 	imes 6 = 36 equally likely rolls, the ones that total 7 are 1-6, 2-5, 3-4, 4-3, 5-2, and 6-1, which is 6 outcomes. So the chance is 6/36 = 1/6. Seven is the single most likely total, which is why it makes a natural jackpot. The bonus averages: 10 	imes 	frac{1}{6} = 	frac{10}{6} = 	frac{5}{3}. Add the two pieces: EV = 7 + 	frac{5}{3} = 	frac{21}{3} + 	frac{5}{3} = 	frac{26}{3} \ap…

Two dice, win the sum plus ten dollars for a seven

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