Variance of a prize wheel's cash payout

Question

A prize wheel pays a cash amount X (in dollars) on each spin: Payout 10 20 30 ------------ Chance 20% 50% 30% What is the variance of the payout?

Accepted Answer

Variance measures how far payouts spread around the average. With larger numbers the squares get big quickly, so keep the arithmetic tidy. Use \mathrm{Var}(X) = E[X^2] - \big(E[X]\big)^2. Mean. E[X] = 10(0.2) + 20(0.5) + 30(0.3) = 2 + 10 + 9 = 21. Average of the squares. Square each payout first: E[X^2] = 100(0.2) + 400(0.5) + 900(0.3) = 20 + 200 + 270 = 490. Subtract. \mathrm{Var}(X) = 490 - 21^2 = 490 - 441 = 49. The standard deviation is \sqrt{49} = 7 dollars, so payouts sit near \21, give or take about \7. Takeaway. Variance carries squared units (here squared dollars), which is why it lo…

Variance of a prize wheel's cash payout

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