Weight each value by its probability to find the expected value. Calculate variance as E(X²) minus the square of E(X), and compare a game's stake with its expected payout to judge mathematical fairness.
At another spinner stall at the fair, the payout X is 0, 1 or 5 pounds with probabilities 0.6, 0.3 and 0.1. A distribution gives both the average payout and its spread. Its probabilities must be nonnegative and total 1. Find a missing probability by subtracting the known probabilities from 1. For the average, multiply each prize by its probability, then add the products.
In 100 independent spins with these fixed chances, the expected counts are 60, 30 and 10. These are not promised counts. The expected total payout is 0 × 60 + 1 × 30 + 5 × 10 = £80, or £0.80 per spin. From the distribution, E(X) = 0 × 0.6 + 1 × 0.3 + 5 × 0.1 = 0.8. E(X) is the expected value: each value multiplied by its probability, then added.
Nobody wins 80p on this spinner: it pays £0, £1 or £5. E(X) is the long-run average over repeated independent spins with fixed probabilities, not a promised payout on one spin. It need not be a value X can take. A probability-weighted mean always lies between the smallest and largest possible values, so a mean outside that range signals a calculation or table error.
Variance is the probability-weighted average of the squared distances from E(X), so it is nonnegative. Use Var(X) = E(X²) - [E(X)]². Square each value, weight by its probability and add for E(X²). For the spinner, E(X²) = 0² × 0.6 + 1² × 0.3 + 5² × 0.1 = 2.8, so Var(X) = 2.8 - 0.8² = 2.16. To recover the mean, [E(X)]² = E(X²) - Var(X). Take its square root and use the stated sign of the mean.
Weight the squares, add, then subtract the square of the mean.
A fair stake equals the expected total payout, including any returned stake. The player's expected net gain is E(X) - stake. For a target gain, stake = E(X) - target gain. Here a £1 stake gives a mean loss of £0.20; over 400 goes the stall's expected margin before costs is £80. Actual totals can differ. Mean and spread can change separately: replace £5 at chance 0.1 by £50 at 0.01 and move the freed chance to £0. E(X) stays £0.80, but variance rises.
E(X) is a probability-weighted mean and need not be a possible payout. Var(X) = E(X²) - [E(X)]² measures spread. A fair stake equals E(X). Expected margin is stake minus E(X).
Lesson, practice questions and flashcards written for Marble.
Readings and practice questions written for Marble. Linked videos by Khan Academy: Mean (expected value) of a discrete random variable, and Variance and standard deviation of a discrete random variable.
Another spinner stall has its distribution on a card: prizes of 0, 1 or 5 pounds, with probabilities 0.6, 0.3 and 0.1. Dee needs to know what one go is worth so she can set a stake before the gates open.
What total payout should the stall expect over 100 spins?
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