Two dice, win double the sum for a ten-dollar fee

Question

You pay 10 to play. You roll two fair six-sided dice and win twice their sum in dollars (so a total of 9 pays 18). On average, how much do you win or lose per play, after the fee?

Accepted Answer

Find the average payout, then subtract the fee. Average sum. Each die averages 3.5, so the two dice average 3.5 + 3.5 = 7. Double it. The payout is twice the sum. Doubling every payout doubles the average, so the average payout is: 2 	imes 7 = 14 	ext{ dollars.} Subtract the fee. You pay 10 every play: EV = 14 - 10 = 4. You gain an average of 4 per play. Notice you did not need the full table of dice totals: once you know the sum averages 7, scaling and shifting are quick arithmetic on that one number. Takeaway. The sum of two dice averages 7. Multiply by the payout factor, subtract any fee…

Two dice, win double the sum for a ten-dollar fee

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