A raffle question.

[Mike Haven]

My local paper tells a story this week where a group held a raffle, and at the drawing the organiser announced they would draw the third prize first.

A ticket-holder stood up and said, no way, draw the first prize first as usual. He said that if he won the third prize on the first draw then he would have no chance to win the first prize.

Everyone in the audience agreed, and they drew the first prize first.

My gut feeling tells me that it doesn't matter two hoots which way it's drawn, but I can't put the maths down on paper, to write to the Editor to explain why.

Could someone help, please?

Let's say there are 100 tickets sold at $1 each and the three prizes are $50, $30 and $20.

Thank you.

[drbst]

Must be a Fool's Day joke. But a good one, I admit.

[Mike Haven]

[img]/images/graemlins/smile.gif[/img] No, regrettably not. We're talking Nor'n Ireland, here.

Anyone, please?

[cliff]

The ordering of the anouncements don't matter, in either case you can view this as the sponsors drawing three names and ordering them from first to third. The order in which the announcements are made don't effect the outcome.

Your expected winnings with 3rd place read first with N particpants and prizes of P3, P2, and P1 dollars and a buyin of B dollars are
(prob of 3rd place) x P3 + (prob of 2nd place)x P2+(prob of first place) x P1 -B
=(1/N) P3 +(prob of not getting 3rd)x(prob of drawing given not 3rd)xP2 +(prob of not getting 2nd or 3rd)x(prob of drawing given not second or third)xP1-B

=(1/N)xP3 +
(N-1)/ (N) x 1/(N-1)P2 +
(N-2/N)x1/(N-2)P1
=3/N(P1+P2+P3)-B

if you reverse the order you get the same outcome (just switch the order of P3, P2, and P1 above).

In essence, choosing three names all at once and then ordering them is the same as choosing names one at a time under a predetermined ordering of winnings.

[WhiteWolf]

OK, I'll try, but it probably won't be as concise as a phzon or BruceZ post.


The chance of winning $50 if the first place prize is drawn first is obviously 1 / 100.

If the tickets are drawn in reverse order (first for the $20, then the $30, and then the $50 prize), in order to win $50 I need to NOT be drawn for the $20 prize (99 / 100 chance), then NOT be drawn for the $30 prize (98 / 99 chance), then be drawn for the $50 prize (1 / 98) chance. Multiplying these:

(99 / 100) * (98 / 99) * (1 / 98) = (1 / 100)

You get the same odds either way.

Another argument, not as rigorous but one that may satisfy those either afraid or skeptical of math: If the guy thinks that drawing the $50 prize first improves his chances of winning that prize, than it must improve the chances for everyone else as well. However, since only one person can win that prize no matter how the tickets are drawn, it should be obvious that the drawing order does not improve anyone's chances.

[Luzion]

Whitewolf: Your first explanation is very clear and simple. Good job. The odds are 1 out of 100 to draw to a specific prize regardless of order.

But Im not too clear on your alternate explanation. [img]/images/graemlins/frown.gif[/img]

[AaronBrown]

Cliff and WhiteWolf gave you excellent answers. I'd add something else.

It doesn't matter to the final result, but it does matter to feelings. Have you ever been in a competition in which winners were announced (as they usually are) in reverse order? When you hear you won third, part of you is happy to have won anything, part of you is unhappy you didn't take first or second.

People emotionally overestimate the chance of winning raffles. The difference between 1 in 10,000 and 1 in 100 million is not that big from a standpoint of personal experience. Only if you say, "winning a 1 in 100 million lottery is like having to win two 1 in 10,000 lotteries in a row" do non-mathematical people start to comprehend the difference.

If they announce the third prize first, part of your brain is hoping you don't win, because it wants first. When you don't win any prizes, another part of your brain will complain that you lost because you weren't hoping to win the third prize. You will feel cheated, you didn't get a fair chance to hope. After all, if you don't believe hope makes a difference, you wouldn't buy raffle tickets.

If they draw from first place down, you can hope your heart out for each prize. 97 people still lose, but they have the satisfaction of knowing they did their best.

[cliff]

I was going to say something along these lines (feelings). A key to lotteries and the like is that people many don't nessesarily accept the basic tenant that such events are random. They believe that luck is a force that can be manipulated and that "strange" actions like announcing winners backwards messes up the natural, pre-ordained, flow of events. For instance, when a dealer starts at the wrong position in a poker game and people demand a redeal because they have the wrong cards. No probabilistic argument will change this desire as they do not believe that things are fundamentally random, rather that they will win when they are due (and messing up the normal flow may cheat them of their due)

[almostbusto]

[ QUOTE ]
I was going to say something along these lines (feelings). A key to lotteries and the like is that people many don't nessesarily accept the basic tenant that such events are random. They believe that luck is a force that can be manipulated and that "strange" actions like announcing winners backwards messes up the natural, pre-ordained, flow of events. For instance, when a dealer starts at the wrong position in a poker game and people demand a redeal because they have the wrong cards. No probabilistic argument will change this desire as they do not believe that things are fundamentally random, rather that they will win when they are due (and messing up the normal flow may cheat them of their due)

[/ QUOTE ] a lot of players get upset because they don't want to be cheated. having rules, routines, and procedure makes it much harder for a game to be crooked.

[cliff]

I understand this completely, and understand the utility with a set of rules that are always followed. I had in mind home games I have been at where people state they want a redeal because they have the wrong cards (particularly low stake games where it is not worth cheating and nobody is competent to cheat). I don't play blackjack but have heard people say similar things about players who hit when they shouldn't (or stand when they should hit), where the other players feel that thy arn't getting the cards they were supposed to due to the offending player messing up the deck. I don't think this would be true of many serious card players, but there is a sense of mojo that I have seen in many games. My main point was that many peopl play games of chance with such a mindset, whether they admit it or not, and arguing that the expected outcome is the same regardless of ordering (in the raffle example) is not likeely to change their worldview.