Raffle Odds Calculator

Table of Contents

• How raffle odds work
• Parts of the calculator
  • Tickets sold, your tickets and prizes
  • Chance of winning and odds
  • Chance of losing
  • Expected prizes
  • Chance per ticket
• Example: three prizes
• Why the quick formula is wrong
• Frequently asked questions

How raffle odds work

In a raffle, every ticket sold goes into the drum, and each prize is won by one ticket drawn at random. A ticket that has been drawn doesn't go back in, so it can win only once. Your chance of winning depends on three numbers: how many tickets were sold (N), how many of them are yours (k), and how many prizes there are (p).

With one prize, it is simply your share of the tickets:

chance = k / N

With several prizes, the easiest way is to work out the chance that none of your tickets is drawn, and take it away from 1. The calculator does this for raffles of any size, up to millions of tickets and thousands of prizes. A tombola, a prize draw or a lottery where each ticket is drawn once works the same way.

Parts of the calculator

Enter the tickets sold. Your tickets and the prizes start at 1, so the chance of a single ticket shows straight away; change them to match your raffle.

Tickets sold, your tickets and prizes

• Tickets sold (N): every ticket in the draw, yours included.
• Your tickets (k): how many of them you hold. It can't be more than the tickets sold.
• Prizes (p): how many prizes there are, so how many tickets will be drawn. With more prizes than tickets the raffle is impossible, and the results stay empty.

Chance of winning and odds

The chance of winning is the chance that at least one of your tickets is drawn, as a percentage. The odds say the same as "1 in …": a chance of 5% is odds of 1 in 20.

They come from the chance of no prize, P(none), which the calculator works out in the background: the number of ways to draw all p prizes from the tickets that aren't yours, divided by the number of ways to draw them from all the tickets. The chance of winning, P(win), is what is left:

• P(none) = C(N - k, p) / C(N, p)
• P(win) = 1 - P(none)
• odds = 1 / P(win)

C counts combinations. When fewer tickets that aren't yours are left than there are prizes, one of yours is certain to be drawn, and the chance is 100%.

Written out, the combinations become factorials:

P(none) = (N - k)! × (N - p)! / ((N - k - p)! × N!)

Factorials grow too large for any calculator after 170!, so in the background the calculator works with their logarithms, which stay small: ln of (N - k)!, ln of (N - k - p)!, ln of N! and ln of (N - p)!. They are exact up to 19!, and from 20! on they come from Stirling's formula, accurate to about 12 decimal places, so a raffle of millions of tickets still gives an answer. With a single prize none of this is needed: the chance of no prize is simply (N - k) / N, so one-prize raffles are exact at any size.

Chance of losing

The chance that none of your tickets is drawn, so you win nothing: 100% - chance of winning.

Expected prizes

The number of prizes you would win on average if the same raffle were run many times:

expected prizes = k * p / N

It can be more than 1 when you hold a large share of the tickets, and it is not the chance of winning: with 10 of 100 tickets and 10 prizes you can expect 1 prize, but you still win nothing about 33% of the time.

Chance per ticket

The chance that one particular ticket wins a prize: p / N.

Example: three prizes

500 tickets were sold, you hold 10 of them, and there are 3 prizes.

1. First draw. 490 of the 500 tickets aren't yours, so the chance it misses you is:

   490 / 500 = 0.98

2. Second draw. One ticket has gone, so 489 of the 499 left aren't yours:

   489 / 499 ≈ 0.97996

3. Third draw. 488 of the 498 left aren't yours:

   488 / 498 ≈ 0.97992

4. Multiply to get the chance that all three draws miss you, and take it from 1:

   0.98 × 0.97996 × 0.97992 ≈ 0.94108

   1 - 0.94108 = 0.05892

Your chance of winning at least one prize is 5.89%, odds of about 1 in 17. With a single prize it would be 10 / 500 = 2%.

Why the quick formula is wrong

A common shortcut works out the chance per ticket, p / N, and treats every ticket as a separate try: 1 - (1 - p/N)^k. That would be right if each drawn ticket went back into the drum. In a raffle it doesn't, so the shortcut always comes out too low, and the more tickets you hold, the more it is off.

The extreme case shows it: in a raffle of 10 tickets with 1 prize, buying all 10 makes you certain to win, but the shortcut gives 1 - 0.9^10 ≈ 65%. This calculator counts the draws as they really happen.

Frequently asked questions

How do you calculate the odds of winning a raffle?

With one prize, divide your tickets by the tickets sold: 5 tickets out of 400 is 5 / 400 = 1.25%, or 1 in 80. With several prizes, multiply the chances that each draw misses you, as in the example above, and take the result from 1.

Does buying more tickets increase my chances?

Yes. With one prize, your chance grows in direct proportion: twice the tickets, twice the chance. With several prizes it grows a little less than that, since two of your tickets can be drawn in the same raffle.

What are my odds with multiple prizes?

Higher than with one prize, but not by the full number of prizes: 10 tickets out of 500 is a 2% chance with 1 prize and 5.89% with 3 prizes, not 6%. Enter the number of prizes, and the calculator works out the exact chance.