School Fair Spinner: Expected Profit and Risk
At a school fair, a stall runs a spinner game. The spinner has four sectors: a red sector of 180 degrees, a blue sector of 90 degrees, a green sector of 45 degrees and a gold sector of 45 degrees. Players pay $2.00 per spin. The prize rules are: red wins nothing; blue wins a $1.00 voucher; green wins a $3.00 voucher; gold wins a $6.00 voucher. The stall has 400 spins available and must raise at least $200.00 for charity after paying prizes. A student says the game is fair because two of the four sectors are winning sectors and the top prize is only three times the cost to play. The stall leader says the expected profit per spin is what matters, but is unsure whether 400 spins is enough to be confident of reaching $200.00.
1. Calculate the probability of each outcome and the expected profit per spin for the stall.
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The spinner totals 360 degrees. P(red) = 180/360 = 1/2 = 0.5. P(blue) = 90/360 = 1/4 = 0.25. P(green) = 45/360 = 1/8 = 0.125. P(gold) = 45/360 = 1/8 = 0.125.
Expected prize per spin = 0 x 0.5 + 1 x 0.25 + 3 x 0.125 + 6 x 0.125 = 0 + 0.25 + 0.375 + 0.75 = $1.375. Expected profit per spin for the stall = income - prizes = $2.00 - $1.375 = $0.625.
The student's reasoning is wrong. Having two winning sectors does not make the game fair, because the sectors are not equal in size and the prizes have different values. Expected value, not the number of winning sectors, decides fairness.
2. Calculate the expected total profit over 400 spins and assess whether the $200 target is likely to be met. Include an approximate measure of variability.
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Expected total profit = 400 x $0.625 = $250.00. That is $50 above the $200 target.
To judge risk, use the profit per spin values: red: +$2 with probability 0.5; blue: +$1 with probability 0.25; green: -$1 with probability 0.125; gold: -$4 with probability 0.125. E[X^2] = 4 x 0.5 + 1 x 0.25 + 1 x 0.125 + 16 x 0.125 = 2 + 0.25 + 0.125 + 2 = 4.375. Variance = 4.375 - 0.625^2 = 3.984375. Standard deviation per spin = sqrt(3.984375) = about 1.996. For 400 spins, standard deviation of total profit = 1.996 x sqrt(400) = 1.996 x 20 = about $39.9.
The target $200 is about 1.25 standard deviations below the mean of $250. Using a normal approximation, the probability of falling below $200 is roughly 10%. So the target is likely but not highly certain. The approximation is rough because the profit distribution is skewed by the -$4 gold outcome, so the stall should not treat $200 as guaranteed.
3. Find the probability that at least one gold prize occurs in five spins, and explain what that probability does and does not show.
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P(gold) = 1/8, so P(not gold) = 7/8. P(no gold in 5 spins) = (7/8)^5 = 16807/32768 = about 0.513. P(at least one gold in 5 spins) = 1 - 0.513 = 0.487, or about 49%.
This shows that a gold prize in five spins is not rare: it happens about half the time in sets of five spins. Therefore, if a gold prize appears early, that alone is not evidence that the spinner is rigged or unfair. The probability does not show that the stall will make a profit in a short run of five spins; expected profit is a long-run idea. A small number of spins can easily produce a loss even when the expected profit per spin is positive.