A fundraising raffle with a suspiciously generous rule
A school raffle sells 400 tickets at $2 each. There are three prizes: first prize $150, second $80, third $40. The organiser adds a rule: anyone who buys five tickets gets one free ticket, and free tickets are entered in the draw but cost nothing. Records show 62 students bought five tickets, 130 bought one ticket, and the rest bought two or three. A parent complains that the raffle is "rigged to lose money" because the prize pool ($270) is less than the ticket revenue. The organiser counters that the free tickets dilute everyone's chance, so buyers are worse off than they think. The class must separate the organiser's revenue question from the buyer's expected-value question.
1. Compute total revenue and the total number of tickets in the draw, then find the probability that a single paid ticket wins any prize.
Read the worked answer
Revenue: 62 students × 5 tickets × $2 = $620, plus 130 × 1 × $2 = $260. The remaining buyers are 400 − 62 − 130 = 208 students buying two or three tickets; without a split we cannot price them exactly, so state the assumption. If we take the stated total of 400 tickets sold as the paid count, revenue is 400 × $2 = $800, but that double-counts the 62 five-ticket buyers already included. Cleaner: treat 400 as the total paid tickets, so revenue = $800.
Free tickets: 62. Total tickets in draw = 400 + 62 = 462.
Probability a given paid ticket wins some prize = 3/462 ≈ 0.00649, about 0.65%.
2. Find the expected net gain for a student who buys five tickets, and compare it with the organiser's claim that free tickets make buyers worse off.
Read the worked answer
Cost = 5 × $2 = $10. This buyer holds 6 tickets (5 paid + 1 free) out of 462.
Expected prize value = (6/462) × (150 + 80 + 40) = (6/462) × 270 ≈ 0.012987 × 270 ≈ $3.51.
Net expected gain ≈ $3.51 − $10 = −$6.49.
The organiser is right that the free ticket dilutes the chance per ticket, but wrong to imply this is the main problem. Even ignoring dilution entirely (6/400), expected prize value is only about $4.05, still far below $10. The raffle is heavily negative for every buyer because the prize pool ($270) is small relative to revenue. The real issue is the payout ratio, not the free-ticket rule.
3. The parent says the raffle "loses money". Identify the error in that statement using your figures.
Read the worked answer
The parent has confused the organiser's position with the buyers'. For the organiser, revenue $800 exceeds prizes $270, so the raffle raises about $530 before costs - it makes money. For a buyer, expected return is roughly $3.51 on a $10 outlay, so the buyer loses. Both statements can be true simultaneously because a raffle is a transfer, not a value-creating activity. Saying the raffle "loses money" without naming whose money is the error.