Market on the Product of Two Dice

Question

I'll roll two fair dice and pay you their product in dollars. Make me a market: state your theo first, then your bid and offer, and justify the width.

Accepted Answer

Theo. Independence factorizes the expectation: mathbbE[XY] = mathbbE[X],mathbbE[Y] = 3.5 times 3.5 = mathbf12.25. No enumeration needed, and saying the word "independence" out loud is most of the credit. Width. The product is wild: it ranges from 1 to 36, and the standard deviation is large, mathbbE[X^2Y^2] = left(mathbbE[X^2]right)^2 = left(tfrac916right)^2 approx 230.0, qquad sigma = sqrt230.0 - 12.25^2 approx 8.9. A one-shot payout with sigma approx 9 deserves a real spread: something like 11.5 bid / 13 offer is defensible, edge on both sides, tight enough to be taken seriously. Quoting 12.2 / 12.3 on a single roll is bravado; quoting 8 / 17 says you didn't compute theo. Shape matters too. The distribution is right-skewed (mean 12.25, but the median outcome is 9 and the modal payouts are small; 36 happens once in 36 rolls). You'll lose small amounts often as a seller and occasionally pay out big, worth saying so the interviewer knows you see past the mean. The trap: computing mathbbE[XY] as mathbbE[X+Y]-style intuition gone wrong (quoting around 7), or asserting mathbbE[X^2] = mathbbE[X]^2 when asked for the variance. Independence factorizes means of products, not squares of sin

Market on the Product of Two Dice

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