Expected product of two four-sided dice

Question

You roll two fair four-sided dice (faces 1, 2, 3, 4) and are paid the product of the faces, in dollars. Solve in your head. What is the fair price of this game? Aim: under 10 seconds.

Accepted Answer

The dice are independent, so the expectation of the product factorises: E[XY] = E[X]\,E[Y]. Each die averages \frac{1 + 2 + 3 + 4}{4} = 2.5, so E[XY] = 2.5 	imes 2.5 = 6.25. Square of 2.5: it ends in 5, so use the ending-in-5 trick, 2 	imes 3 = 6, append 25, giving 6.25. The brute-force route sums all 16 products and divides by 16; the structure gives it instantly. Takeaway. E[XY] = E[X]E[Y] holds only under independence, say the word out loud. Contrast the sum, E[X + Y] = 5, which needs no independence at all.

Expected product of two four-sided dice

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