Expected product of two dice

Question

You roll two fair dice 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

Independence means the expectation factorises: E[XY] = E[X]\,E[Y] = 3.5 	imes 3.5 = 12.25. Mental square of 3.5: it ends in 5, so use the ending-in-5 trick, 3 	imes 4 = 12, append 25: 12.25. The slow route, summing 36 products, takes minutes; the structure takes seconds. Takeaway. E[XY] = E[X]E[Y] only under independence, say that word out loud in the interview. Contrast with the sum, E[X+Y] = 7, which needs no independence at all. Follow-up they may ask: E[X^2] = \frac{1+4+9+16+25+36}{6} = \frac{91}{6} \approx 15.17 
e 12.25, precisely because X is not independent of itself.

Expected product of two dice

Your answer

More Mental Math questions