You win the product of two dice

Question

A game pays you X \cdot Y dollars, where X and Y are two independent fair six-sided dice. What is the fair (zero-EV) price to play? Contrast it with the single-die game that pays X^2, and make a market.

Accepted Answer

For independent X and Y, expectations of products factor: E[XY] = E[X]\,E[Y]. Each die averages 3.5, so E[XY] = 3.5 	imes 3.5 = \12.25. Here is the instructive contrast. The game that pays X^2 (one die, squared) is worth E[X^2] = 91/6 \approx \15.17, strictly more than 12.25. Both look like "roughly 3.5 	imes 3.5," but they differ by the variance of a single die: E[X^2] = (E[X])^2 + \mathrm{Var}(X) = 12.25 + 35/12 \approx 15.17, whereas E[XY] = (E[X])^2 exactly because two independent draws carry no shared variance to inflate the product. The lesson: X \cdot X is not the same bet as X \cdot…

You win the product of two dice

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