Why raffles are easier to calculate than lotteries
A raffle or prize draw is one of the few chance games where the odds are usually simple to work out. Unlike a lottery, where you pick numbers from a huge range, a raffle draws winning tickets from the tickets actually sold. If you know how many tickets exist and how many you hold, you can work out your chance directly.
This article explains the arithmetic with invented numbers. Real raffles have their own rules, which should be checked, and in some places they are regulated.
One prize: tickets held over tickets sold
With a single prize drawn at random from all the tickets, each ticket has the same chance of winning. If you hold t tickets out of N sold, your chance is t ÷ N.
For example, suppose 500 tickets are sold and you hold 5. Your chance of winning the single prize is 5 ÷ 500 = 0.01, or 1 per cent. That is the same as 1 in 100. Whoever holds the other 495 tickets collectively has the remaining 99 per cent.
The same formula shows why the number sold matters. If the raffle sold 1,000 tickets rather than 500, the same five tickets would give 5 ÷ 1,000 = 0.5 per cent. Your chance halves because the pool doubled, and you cannot know the final pool size in advance if tickets are still on sale.
Buying more tickets
Holding more tickets raises your chance, but in proportion to the pool and only gradually. Going from 5 to 10 tickets out of 500 raises your chance from 1 per cent to 2 per cent. Note that this slightly overstates the effect if your extra tickets also enlarge the pool. With 495 other tickets and your own 10, the pool is 505, so your chance is 10 ÷ 505, about 1.98 per cent. Your chance is still very modest, and the cost has doubled.
Each ticket also costs money, so what matters is how the chance compares with the price, which is where expected value comes in. If a 5 ticket gives you a 1 in 500 chance of a single 200 prize, the average return per ticket is 200 ÷ 500 = 0.40, well below 5. For many raffles, though, the purpose is to raise money for a cause rather than to give a fair return, and people buy tickets to support the cause. That is a separate matter from the odds.
Several prizes
If a raffle has more than one prize, there are two common set-ups. In the first, a ticket that has won is removed from the drum, so no ticket wins twice. In the second, tickets are returned and could win more than once. Most small raffles use the first, and the rest of this section assumes it.
Suppose there are 500 tickets, three prizes, and you hold 5 tickets. The easiest way to find the chance of winning at least one prize is to find the chance of winning none and subtract from 1.
- The chance your tickets all miss the first prize is 495 ÷ 500 = 0.99.
- Given that, the chance they miss the second is 494 ÷ 499, which is about 0.98998.
- Then they must miss the third: 493 ÷ 498, about 0.98996.
Multiplying gives 0.99 × 0.98998 × 0.98996, which is about 0.9702. So the chance of winning none is about 97.02 per cent, and the chance of winning at least one prize is about 1 − 0.9702 = 0.0298, or roughly 2.98 per cent.
A rougher shortcut gives a similar answer: three prizes times a 1 per cent chance each is 3 per cent. The shortcut is slightly high because it ignores the small chance of winning more than one prize, and because the exact calculation accounts for each draw removing a ticket. For small holdings in a big pool the shortcut is close enough.
The expected number of prizes you win is exactly 3 × 5 ÷ 500 = 0.03. Expected value works the same way as for any other game of chance: multiply the chance by the prize, and add up across prizes.
Rules that change the calculation
- One prize per person. If a person cannot win twice, a person holding many tickets has a slightly lower chance of a second prize than the simple fraction suggests, and the calculation changes.
- Prizes of different value. The simple chance of winning something does not tell you the chance of winning the best prize, which is the single-prize formula applied to the first draw.
- Tickets still on sale. Until sales close, the pool is unknown, so any calculation is an estimate based on a guessed total.
- Unsold tickets. If the draw uses a fixed number of numbered tickets and some are unsold, the chance per sold ticket is different from the chance per printed ticket.
- Free entry or limited entries. Prize draws that allow one entry per person turn the calculation into one over the number of entrants.
Putting the numbers in perspective
Raffle odds can look far better than lottery odds. A chance of 1 in 100 or 1 in 500 is vastly more likely than 1 in 14 million. The scale is simply different, and it also means the prizes are far smaller. The method, though, is the same: counting what could happen, finding the share that is favourable, and comparing the chance with the price of a ticket.
As with any game of chance, past draws do not affect future ones, and a ticket bought in the hope of recovering earlier spending is a poor reason to play. Spend only an amount you would be comfortable giving away.