Expected product of three dice

Question

You roll three fair dice and are paid the product of all three faces, in dollars. Solve in your head. What is the fair price of this game? Aim: under 15 seconds.

Accepted Answer

Independence factorises the expectation across all three: E[XYZ] = E[X]\,E[Y]\,E[Z] = 3.5^3. Build it in two steps: 3.5^2 = 12.25 (ends in 5: 3 	imes 4 = 12, append 25), then 12.25 	imes 3.5 = 12.25 	imes 3 + 12.25 	imes 0.5 = 36.75 + 6.125 = 42.875. So the fair price is \42.875. Takeaway. Independence extends to any number of factors: E[X1 \cdots Xk] = \prod E[Xi]. Contrast the sum, E[X + Y + Z] = 10.5, which needs no independence, the factorisation is what makes the product tractable.

Expected product of three dice

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