# Odds of winning multiple raffle prizes

A few weeks ago, my wife and I went to a fundraising banquet. We bought 12 raffle tickets, of what I estimate was about 4,000 sold. There were about 30 prizes drawn (tickets pulled randomly from a big bowl), and — to everyone’s surprise — we won four prizes. The prizes were drawn over the course of the evening (ca. 8 draws of 4 prizes each, separated by about 15 minutes), and the bowl was shaken up each time. We didn’t win two prizes in any single draw (so our tickets weren’t “clumped”, as suggested in a comment below).

Question #1: Am I correct that the odds of winning the first prize was roughly 12/3995 (because we didn’t win the very first draw), the second was about 11/3990, the third was about 10/3980, and the last was about 9/3970?

Question #2: To figure out the odds that we’d win four prizes (n.b. nobody else in the room won more than 1!), do I multiply those individual odds?

Question #3: Do the other (ca. 26) draws have to be considered in calculating our odds over the whole evening?

• It's more than one in a million--and "Million-to-one chances crop up nine out of ten times.". However, when such events occur in lotteries, it's usually a safe bet the drawing wasn't completely random: your tickets might have clumped and been drawn out as a clump. – whuber Dec 5 '17 at 16:50
• @whuber: Not clumped (see edited question). And, to anticipate your possible next suggestion/comment: each of the eight draws (except the last one, which they made me do!) was done by a different child “assistant” from the audience — so there wasn’t a "fix" in. =) – Kieren MacMillan Dec 5 '17 at 17:18
• That's not evidence about clumping! It's extremely difficult to mix a bowl of papers thoroughly. Without such mixing, it doesn't matter whether different people draw tickets. I'm not suggesting any cheating: I'm only saying that this was extremely likely not to have been a good randomizing device, and your results are pretty strong evidence of that. BTW, in what sense are "30 prizes drawn" (from the question) and "each of the eight draws" (from your comment) descriptions of the same process? – whuber Dec 5 '17 at 17:38
• @whuber: All the tickets were purchased, ripped apart, and put in the bowl at the beginning of the evening. Over the course of the evening, there were 7 or 8 draws (I forget exactly), one after each round of trivia. In each draw, 3-5 tickets were drawn (1 at a time), with a single prize given away for each ticket drawn. So 7 (or 8) draws x 3-5 tickets per draw = 30ish prizes given out. Sorry I didn't keep track exactly at the time. Sure seems pretty random to me… – Kieren MacMillan Dec 5 '17 at 17:46
• That mechanism alone can introduce such correlations in results; it is manifest the draws were not physically independent. – whuber Dec 5 '17 at 17:48

Ignoring issues like

• how the tickets are clumped, which is hard to address without getting into empirical rather than purely mathematical matters, and
• the probability of this happening to any of the participants, instead of just you, which is hard to address without knowing how many tickets each person bought,

we can consider an urn model with 4,000 balls, 12 of which are white. If you draw 30 balls without replacement, what's the probability of getting at least 4 white? This can be computed by evaluating the CDF of the appropriate hypergeometric distribution at 3 and subtracting this from 1. In R,

> 1 / phyper(3, m = 12, n = 4000 - 12, k = 30, lower.tail = F)
[1] 818603.1


So the probability is about 1 in 800,000.

• +1. It's also helpful to extend this analysis to consider very simple models of clumping. For instance, the chance that one clump of 4 united tickets would be withdrawn in a sample that takes 8 clumps out of a bowl comprised of 999 randomly assembled clumps plus the united tickets is 8/1000 = 1/125. This shows that a clumping hypothesis still leads to a relatively small chance of withdrawing the clump--but it's almost four orders of magnitude greater. That helps us gain a better perspective concerning what might have happened. – whuber Dec 5 '17 at 18:40
• All tickets were separated — there were no united tickets in any of the 4000. – Kieren MacMillan Dec 5 '17 at 18:49