Two dice, win the sum but lose four on snake eyes

Question

You roll two fair six-sided dice. You win the sum in dollars, except if you roll snake eyes (both dice show 1) you instead lose $4 (and collect nothing from the sum).

On average, how much do you win or lose per roll?

Accepted Answer

The cleanest way is to add up the payout over all 36 equally likely rolls, then divide by 36. Step one: the total of all the sums. If you simply won the sum on every roll, the average would be 3.5 + 3.5 = 7. Since there are 36 rolls, the sums across all rolls add up to 36 	imes 7 = 252. Step two: remove the snake-eyes sum. Snake eyes is the single roll 1-1, whose sum is 2. But on that roll you do not win the sum, so take those 2 dollars out of the total: 252 - 2 = 250. Step three: add the snake-eyes penalty. On that same roll you instead lose 4, so add negative four: 250 + (-4) = 246. Step f…

Two dice, win the sum but lose four on snake eyes

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