Expected product of two mismatched spinners

Question

You spin two independent spinners: one uniform on $\{1, \dots, 10\}$ , the other uniform on $\{1, \dots, 6\}$ . You are paid the product, in dollars. Solve in your head.

What is the fair price of this game? Aim: under 10 seconds.

Accepted Answer

Independence lets the product's expectation factorise: E[XY] = E[X]\,E[Y]. A uniform spinner on 1 to n averages \frac{n+1}{2}, so the first averages \frac{11}{2} = 5.5 and the second \frac{7}{2} = 3.5. Then E[XY] = 5.5 	imes 3.5 = 19.25. Mental 5.5 	imes 3.5: 5 	imes 3.5 = 17.5, plus 0.5 	imes 3.5 = 1.75, total 19.25. Takeaway. For a uniform spinner on 1 to n, the mean is \frac{n+1}{2}; under independence just multiply the two means. The two variables need not share a range or a scale, only independence.

Expected product of two mismatched spinners

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