Raffle Probability Question

Factual Questions
1

My workplace organises a raffle in conjunction with our annual Xmas party, with 4-5 prizes on offer. We have ~200 employees who for the sake of the exercise we’ll say buy 5 tickets each.

At the 2018 party the raffle prizes were drawn in order from 1st to 5th, which to me flies in the face of any dramatic build up, and every other raffle I’ve ever seen drawn. I found out today by random discussion that the reason for it was so that if someone won a lesser prize, their chances of winning 1st prize wasn’t lessened.

It sounds wrong to me, but I’m not sure.
I can see how it would be easy to view as you have one less ticket in the draw, (Any minor winning tickets are not replaced in the barrel) so your chances have decreased by 1/1,000 for subsequent draws. But I suspect that from a probability standpoint that is too simple of an approach.

2

Assuming 200 people with 5 tickets each, the chance of you winning the prize which is drawn first is 5/(5*200) = 1/200

The chance of you winning the prize which is drawn next as well is (5-1)/(1000-1) = 4/999 ~ 1/250.

So there is quite a difference in the chance to win the first prize for those who have already won any of the prizes before if you do it the classic way.

Purely from a fairness perspective, I think it is completely ok if your chance of winning first prize decreases if you already won a prize (except if prizes 2-5 are a 10 dollar gift card and 1st prize is a cruise or something like that)

3

Your instinct is correct. The reasoning they provided is wrong. It’s also a very common mistake, so it’s not surprising.

Basically, while it’s true that after the first draw, the probabilities change, before any drawing takes place, the order of the draw makes no difference probabilistically.

It’s probably easiest to demonstrate via examples. Let’s construct a very simple example - there are only two people (Person A and Person B) and each has two tickets and there are two prizes. As in your example, let’s say the 2nd prize is awarded first.

There are really only 4 unique results here:

1st prize - Person A, 2nd prize - Person A
1st prize - Person A, 2nd prize - Person B
1st prize - Person B, 2nd prize - Person A
1st prize - Person B, 2nd prize - Person B

Let’s dive into the numbers a bit.

If Person A wins the 1st prize (1/2 the time), there are 3 tickets left. Person A has a 1/3 chance of winning the 2nd prize (the second ticket was already used).

But if Person B wins the 1st prize (1/2 the time), Person A now has a 2/3 chance of winning the 2nd prize.

So, 1/2 of the time after the first draw, Person A has only a 1/3 chance of getting the 2nd prize, but the other 1/2 of the time after the first draw, Person A has a bigger 2/3 chance of getting the 2nd prize.

Those two offset.

The same thing happens in your case. The chances of somebody winning the 1st prize may be reduced if they win lesser prizes first, but that’s offset by the increased chance if somebody else wins the lesser prizes. Those chances offset each other.

That said, and as you note, while there may not be a mathematical difference there is a psychological difference to prefer drawing the top prize last.

4

I’ll note that there’s an obvious way for the lottery organizers to “have their cake and eat it too” here:

Draw five tickets and label them #1 to #5 in order, without disclosing the names of the winners. Then disclose the names in the reverse order, from #5 to #1.

5

As usual when talking about probabilities the questions was answered with two different answers which are both correct.

It depends on what exactly is meant with “chance isn’t lessened”:

  • If it means that everyone should have the same chance of winning the first prize BEFORE the raffle starts, then Great Antibob’s answer is correct, there is no difference which prize gets drawn first and which last
  • If it means that everyone should have the same chance of winning the first prize AT THE TIME that first prize is drawn irrespective of previous wins, then you need to draw the first prize first
6

nm

7

The obvious way, actually - is to put the ticket back in the pot and shake it up again after announcing winner #5, then #4, etc. Your odds don’t go down for further prizes for winning any prize.

I suppose this is fair, as mentioned, if the first prize is far, far more lucrative. If necessary, then announce that if you are drawn for two lesser prizes - draw again…

Or put the drawn tickets back in when it’s time to draw for 1st prize.

8

Exactly. If you do the draws and then announce thing, the people who don’t understand probability will (hopefully) be mollified.

After a few raffles then you can switch to draw the tickets in order 5 to 1 and then announce. And then after a few more go completely with draw/announce alternating.

9

This thread is very pertinent to me as our office also has a regular lottery and usually it is drawn from lowest value prize to highest. A couple of times I have asked for this order to be reversed to make it fairer, and have been told that they prefer to increase the dramatic tension in the draw. They did not attempt to refute my argument as The Great Antibob has done. So, put me down as one of those who doesn’t understand probability, I guess (I have often acknowledged this in the past).

Still, although I follow the mathematics of it, how can I get over the nagging feeling that if my ticket is the first one drawn, at odds of (say) 1/500, why shouldn’t I get the top prize, as opposed to the last ticket drawn winning the first prize having ‘only’ beaten odds of 1/491 (if there are 10 prizes). I know the mathematical difference is small, but it is there. Would it be correct to think of it as that 10th ticket has not only beaten odds of 1/491, but also odds of ((499/500)(498/500)(497/500)…(492/500)) in NOT being drawn previously, which presumably multiplies out to less than the 1/500 odds on the first ticket drawn?

10

I haven’t taken the time to carefully check the math, but I think that’s right.

But you could take ftg’s suggestion and reveal the tickets in the reverse order that they’re drawn in.

11

More or less but the actual probabilities are a bit different and they all work out to the same probability of 1/500.

It balances out in the end that BEFORE any drawing, the probability your ticket will be the 10th one selected is 1/500. The probability your ticket will be the 6th one selected is also 1/500. The probability your ticket will be the 1st one selected is also 1/500. They’re all the same. To be the 10th ticket, it has to “win” at not being drawn the first 9th times and that probability has to be taken into account.

The math:

The probability your ticket is drawn first is 1/500.

The probability your ticket is drawn exactly second is (499/500) * (1/499) = 1/500. The 499/500 is the chance your ticket is NOT drawn first multiplied by the probability it IS drawn 2nd (when there are only 499 tickets left).

The probability your ticket is drawn exactly third is (499/500)(498/499)(1/498) = 1/500. The 499/500 is again the chance your ticket is NOT drawn first multiplied by the probability it is NOT drawn 2nd out of the remaining 499 tickets multiplied by the probability it IS drawn third out of the remaining 498 tickets.

We can continue this way to the 10th ticket:

(499/500)(498/499)(497/498)(496/497)(495/496)(494/495)(493/494)(492/493)(491/492)*(1/491) = 1/500

All that stuff up to (499/500)…(491/492) = 491/500. So there’s a 491/500 chance your ticket will NOT be selected in the first 9 drawings, which is a necessary condition to have any chance to be selected 10th, where there’s one chance out of the remaining 491 tickets. And (491/500)*(1/491) = 1/500.

The TL;DR version is your ticket has the same chance of being drawn 10th as it has of being drawn first. The order in which prizes are given out doesn’t really matter mathematically - the odds are the same for all the drawings - BEFORE any drawings take place. Clearly, as mentioned above, after a few prizes have been rewarded, the odds will shift.

12

Thank you both. I will email the lottery co-ordinator and withdraw my suggestion forthwith, since it all comes out the same - may as well keep the dramatic tension! And if anyone complains that they were ‘unlucky’ to be the first ticket drawn and thus win the lowest prize, I will now be able to explain that it is not so.

We could, but that would make the whole process longer for no mathematical benefit. The draw is done ‘live’ with most ticketholders watching.

13

With the drawing being done with lowest value to highest you could always give the winner the option of keeping the prize or having their ticket thrown back into the drawing. Add some fun and tension “Shirley, you’ve won the thermos! Do you want to keep it or have your ticket thrown back into the drawing for the bigger prizes?”

14

On the other hand, if you do the high-value prize first, then you can give the winner the option to take one of the lower-valued prizes, if it’s something they’d want more. Not everyone will assign the same values to the prizes.

15

(Note that Xema first posted the core idea.)

Longer? I don’t see how drawing all the tickets then doing the announcements makes it longer. It might even make it shorter.

As to “for no mathematical benefit” that’s the point!

It all comes down to psychology. Nothing done during the drawing changes the pre-draw chances of winning any prize. People want to think that something’s changed. That’s all.

It’s like a horse race. The fact that your horse is ten lengths behind at the last turn doesn’t change the original odds. The outcome of the race, the odds and the wager are all that really matter.

My post was just a way to inch people from one style to another.

16

Like I said - put the tickets in after they were drawn, odds stay the same. If someone wins a lesser prize and the big prize (or two lesser prizes) then say “you’ve been upgraded” and draw again for the lower-value prize if you want to avoid griping about someone winning two prizes (which, of course, could happen anyway if they have multiple tickets in the draw).

17

Given that there was about a 1 in 200 chance that the a person would win any prize, I have a hard time feeling sorry for the guy who won the 5th place prize, and so discovered he reduced his chances of winning the grand prize.

Would he really be willing to turn in that prize for another random raffle ticket?