Variance of a prize wheel's cash payout

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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?

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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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