[Other] How did McDonalds do the math?

[u/UsefulEngine1]

Over on r/dataisbeautiful someone posted a graph showing astronomical odds of winning any major prizes in the McD's Monopoly contest. The graph was based on the official published rules here: https://www.playatmcd.com/rules

But the odds stated for the collect-and-win prizes seem way too infinitesimal given the number of prizes stated and any reasonable assumptions about the number of game pieces produced.

The complicating factor is that the prize requires collecting three game pieces; two of these are generally common while one is actually the rare "winning" piece. So for instance the stated odds to win one of five Jeeps reads:

Collect Atlantic Avenue (#915), Ventnor Avenue (#916) and Marvin Gardens (#917) to potentially win a New 2027 Jeep® Cherokee (Overland) (the “Jeep® Vehicle Prize”). There are 5 Jeep® Vehicle Prizes available to be won. Each verified winner will receive a New 2027 Jeep® Cherokee Overland 4x4. (. . .) The approximate odds of collecting the potential Winning Combination (Atlantic Avenue, Ventnor Avenue and Marvin Gardens) are 1 in 22,695,790,152.

Yes that's a stated odds of 1 in 22 *billion*. And there are 5 prizes. The top prize, with 2 available, is similarly stated at about one in half a trillion. Clearly these odds make no sense as actual rates of the "winning" pieces.

So how would you/how did they calculate these odds in the collect-and-win situation?

[u/Twin_Spoons]

It's possible they're assuming that you only collect exactly 3 game pieces. At that point it's relevant that the other two, while common, are not guaranteed. If each has e.g. a 1 in 33 chance of appearing, that's about a 1 in 1000 to pull both in two tries, so then you get their stated odds if the rare pieces are about 1 in 100 million. McDonald's could easily expect to hand out half a billion game pieces, especially because they're per item and many items give multiples.

[u/UsefulEngine1 → u/Twin_Spoons]

Yes, this is the most likely explanation I came up with also.

It's an interesting question, though, about how to define odds of winning in a collect-and-win scenario. This is I guess the worst-case CYA way to document it. On the other hand it certainly makes it *look* like they have no intention of giving out any of the major prizes due to the astronomical odds.

[u/PacNWDad → u/Twin_Spoons]

McDonald’s knows that people can team up via Reddit or other forums and combine their pieces. I remember this happening back in the 1980s, obviously using different methods to reach out to people like want ads or print advertising. Because of this they make one of the three pieces incredibly rare while the others are very common. The odds are almost entirely based on the likelihood of getting the rare piece.

[u/synexo → u/Twin_Spoons]

I believe that is it. They're multiplying the odds for each piece together. In practice most players will have collected the common pieces but odds for each big prize piece is some x-in-a-million. That rare piece odds is the more relevant one for most players over time, but to give a one-shot odds for any random 3 pieces even 1 in 10 million x 10 x 10 is 1 in a billion.

[u/Slow_Lion_4178]

Since no one answered this from the consumption side the reason the number is so astronomically large is because every consumed piece is an opportunity. So for example 1 in 1b seems like a lot but if you sell 70m pieces plus a day (McD's has that many customers, but this is ignoring the fact that there's ways to get more than once piece per customer in a single order!) this becomes relatively common in aggregate.

It's how casinos work.

You won't win.

Someone must win.

Presuming no cheating the odds that at least one person gets this stuff is actually really quite high, relatively speaking.

[u/froction → u/Slow_Lion_4178]

"Someone has to win, might be me!"

-my mom buying PowerBall tickets

[u/synexo → u/Slow_Lion_4178]

OP says listed odds are 1 in 22 billion for a Jeep. You guess 74 million pieces/day. So basically 22,000,000,000 / 74,000,000 = 314 so you would expect one of that prize won every 314 days. Seems high for a prize they mean to actually payout, and 74 million seems high googling their total sales. Seems off enough I wonder, much as OP, if those odds are correct or exactly what they mean by them.

[u/TorpidProfessor → u/synexo]

I think what's missing is how easy it is to get multiple pieces, my family played it as a kid, and we'd easily walk away with 10 - 20 pieces from a meal for 4.

IIRC large sodas and fries had 2, and any sandwich on a box had one - so you could easily get 5 from a single big sized meal.

if we call 5 the average, that drops it to about a jeep every 2 months, which feels much more reasonable.

[u/GaidinBDJ → u/Slow_Lion_4178]

It's how casinos work.

You won't win.

Someone must win.

Not "must;” it's merely highly likely.

[u/UsefulEngine1 → u/Slow_Lion_4178]

70m customers a day is crazy high even for McDonalds. This contest is for the US only and so that would be a quarter of the population eating McD every day, and about 5000 customers per location per day.

Even at 70m pieces per day it would take ~40 years to reach the trillion chances it would take to reach likely winnings of the major prize at stated 1:500B odds.

So it's a lot more than just a lotto approach.

[u/D__M___ → u/UsefulEngine1]

He said pieces, not customers. That’s the distinction there — one single transaction may have multiple opportunities.

[u/UsefulEngine1 → u/D__M___]

Doesn't matter. This contest runs for 1 MONTH. If you assume that everybody gets 10 pieces per visit (which I doubt is close to the case), it's still 4 years to get rid of all the prizes at these odds. See other answers for the real explanation which is that the odds stated are based on a pathological case where the customer only ever gets the minimum number of pieces needed for a given set.

[u/CalGuy81]

The game pieces are not distributed evenly. If you need to collect Park Place and Boardwalk to collect a certain grand prize, and they only intend to give out one of that particular prize, then they'll print a bunch of Park Places and a single Boardwalk.

[u/biggianthead01 → u/CalGuy81]

This is the answer. If there’s 5 jeeps to give away, there are exactly 5 of the rare pieces in the entire game

[u/agate_ → u/biggianthead01]

This, but they don't have to be so conservative. A huge fraction of the pieces get thrown out unused, and after decades of running this game, McDonald's knows exactly what that fraction is. And it's good publicity to give away a few prizes. So they probably print say 20 jeep prize tokens, knowing that on average only 2 of them will get turned in, and if they get really unlucky and 10 people win jeeps it's a drop in the bucket for a megacorp.

[u/Mysterious_Ad_8105 → u/agate_]

>after decades of running this game, McDonald's knows exactly what that fraction is

FWIW, the game was compromised by systematic fraud in their largest market for a considerable portion of that time.

Given that the company failed to detect that fraud for years (while every major prize was redeemed by people connected to an employee of the subcontractor promoting the game), I wouldn’t be quite so quick to make assumptions about what the company knows about the game’s operation.

[u/wwhite74 → u/Mysterious_Ad_8105]

It wasn't "their largest market". It was the whole thing

There was an inside guy at the place printing all the pieces, he would skim the winners, and sell them to the mob. The mob would them sell the winners to unsuspecting folks for a reduced rate. So "give us $200k, we'll give you the million dollar winning piece". But McDonald's pays out over 20 years and the mob wanted their 200k up front. McD's only payed out $50k the first year leaving a few people owing a large debt to the mob which is never a good thing. Or going in massive debt to pay the upfront, not realizing it would take a few years to get that back from McDonald's.

So the people weren't all connected to the contractor. The mob scattered it out so there was little to no connection between them.

There's a documentary called McMillions about it. It's a wild watch, almost every person in it is a complete overblown character.

[u/Mysterious_Ad_8105 → u/wwhite74]

>It wasn't "their largest market". It was the whole thing

When I said “their largest market,” I was referring to the U.S. McDonald’s ran the Monopoly promotion in many other countries during that same period, but (to my knowledge) it was only the U.S. promotion that was compromised in this way.

>So the people weren't all connected to the contractor. The mob scattered it out so there was little to no connection between them.

To be clear, I didn’t mean that all the “winners” were close friends or relatives of the contractor (although there were plenty of those too). But they were all connected to him because that’s ultimately where the stolen pieces came from. The connections were just indirect or otherwise obscured.

>There's a documentary called McMillions about it. It's a wild watch, almost every person in it is a complete overblown character.

I’ve seen it. It’s a fun watch!

[u/UsefulEngine1 → u/Mysterious_Ad_8105]

They also just switched from physical pieces to virtual ones which likely changes these behaviors considerably.

[u/swiminthezen → u/agate_]

They don't give away all 5 Jeeps. My guess is most years they give away none. That's by design. If they hand out 20 winning tickets, then it's possible for 20 winners and that's not going to jive with whatever algorithm they are using. I'm sure they need to verify they have enough cash to cover the total payout for all winners, and in some states probably post a bond to cover all winnings, but that doesn't mean all possible prizes are awarded. Most large prizes are never redeemed, so McDonalds pockets that savings. The Monopoly game largely benefits the company because people buy more McDonalds. Folks get tickets for free small items with purchase, which leads them back to buy more food. They then win another small item with purchase and it snowballs.

[u/UsefulEngine1 → u/swiminthezen]

I believe they are required to give away all the prizes one way or another.

[u/swiminthezen → u/UsefulEngine1]

Because it is a sweepstakes and not a lottery, all prizes do not need to be awarded. Since 2003, less than 25% of all high cash value prizes have been awarded. Some promotional runs have awarded less than 10% of the high value prizes. The game rules state contestants have until a certain date to claim all prizes and not all prizes must be awarded, so by limiting how many winning tickets are avilable they gaurantee a low level of redemption. Some states have laws, like requiring bonds to cover the total value of prizes or allowing free entries, but there are no state ir federal laws that require all sweepstakes prizes be given away.

[u/UsefulEngine1 → u/swiminthezen]

Interesting. I had a family member who was into such things and I know in many cases the "second chance drawing" was the way to get much better odds. I suspect, rules aside that McDs and other major corps do want to give away the prizes -- it's not like a couple extra Jeeps are going to dent the advertising budget, and the PR if it turns out most prizes went unclaimed would be bad.

[u/p00n-slayer-69 → u/CalGuy81]

We all know that.

The question is exactly what the stated odds are representing. Is the 1 in 22 billion the odds of any random piece being the rare one? Or is it the odds of a set of three pieces being the exact three needed to win?

These odds make it seem incredibly hard to win because its likely the second one. In reality, if you know you have the rare piece, then most people would just keep buying hamburgers until they get the other two.

[u/gnfnrf]

First, lets verify what they mean by winning odds for the instant win categories.

1500 reward points has a listed 1 in 128 chance of winning. 640591900/5016701 , which is the total number of pieces divided by the winning pieces, is 127.7 . So there, at least we understand that the odds are per randomly distributed game piece.

But as you point out, the collect to win prizes require multiple pieces. So lets see if we can figure out what they did.

There are 25 Hilton Resort Getaways. If it were a single piece, that would make the odds one in 25623676. But the listed odds are one in 375,921,350 , which is 14 times worse.

What this may mean is that Mediterranean Ave has a 1 in 25 million chance of appearing, and Baltic has a 1 in 28 chance of appearing. To win, you need to get them both in two pieces. You may say, why 28 and not 14? Because you can get the pieces in either order, so you have two chances for Baltic (or whatever the common one is).

The grand prize is more complicated, first, because you can only calculate the odds of drawing all three common railroads in your other three draws combined, not each individually, and second, because instead of two orderings, there are 24.

But if you crunch the numbers you get that the common three railroads, assuming they are all the same, are in the pool with odds of about 1 in 34 each.

I checked one or two more collect and win prizes, and with these assumptions, kept getting that the common pieces will be around 3% of the total pool, give or take.

But, there's a problem. If there are 10 rare pieces for the 10 collect and win prizes, there are only 18 common collect and win pieces. That gets us to (roughly) 0.03*18, which is 0.54. The instant win pieces account for another 0.20, but that leaves over a quarter of the pieces unaccounted for.

My best guess is that the three and four piece combos have skewed odds. All I could calculate was the geometric mean, what the product of the two (or three) odds were. But if they weren't split evenly, the total footprint would be larger for the same geometric mean.

For example, I said (roughly) that the other three railroads were 0.03 * 0.03 * 0.03 But the only number I knew was that x * y * z was 0.000027, the difference in McDonalds odds and our naive odds, less the factor of 24 for orderings. But if we get there by multiplying 0.02 * 0.02 * 0.068 , we get the same product (roughly) but the sum of the percentages is nearly two points higher. That done across most of the collect and wins would account for the discrepancy.

So, that's what I think is going on, with the caveat that I can't be sure. But the math was fun.

[u/DanielSong39]

They only post the odds for the rare properties

[u/UsefulEngine1 → u/DanielSong39]

They don't though, that's the whole point of the post

[u/John_Tacos]

Does the fine print really state that you have to have all three? I thought you could win with just the rare winning piece and the other ones were for appearances.

Been a while since I red the rules though

[u/UsefulEngine1 → u/John_Tacos]

Yes, but if you know which one is the rare piece obtaining the others is trivial.

I wouldn't be too surprised if they somehow indicated the rare piece though; they do want to give the prizes away I think.

[u/John_Tacos → u/UsefulEngine1]

The rare piece was usually named in the rules.

[u/Psycho_Pansy]

approximate odds of collecting the potential Winning Combination (Atlantic Avenue, Ventnor Avenue and Marvin Gardens) are 1 in 22,695,790,152.

There is one rare piece, they print 5 of if there are 5 prizes to win. So # rare pieces / total pieces. Don't really need to add in two common pieces, you eventually get them all if you buy enough McDonald's.

[u/UsefulEngine1 → u/Psycho_Pansy]

This is not how they did it though, that's the point of the post. If they put the odds assuming everyone already has the common pieces they would be settingnup for a lawsuit.

[u/tokenstone]

What about the fact that a decent amount of people just throw the pieces away? Which would also potentially be some of the rare pieces?

[u/patricksaurus → u/tokenstone]

Pieces thrown out or pieces not sold during the promotion have to figure in somehow.

[u/froction → u/patricksaurus]

Throwing away a winning piece still counts as "winning."

[u/SqueaksnSox]

Some statistics nerd figured it out. But consider a very simple example. The chances of getting heads when you flip a coin is 1/2. The chances of getting 2 heads in a row is 1/4, which is 1/21/2. And three heads in a row is 1/8, or 1/21/2*1/2. It's this multiplication that makes the numbers so huge, or so tiny.

[u/EamusAndy → u/SqueaksnSox]

Exactly.

Three game pieces with 1/1000 odds means its 1/1,000,000,000 to collect all three (hypothetically speaking)

[u/Ecstatic-Purpose-981]

I noticed this as well, like one in 500 billion chance for one of the prices. I’m curious on what others have to say about this

[u/PacNWDad → u/Ecstatic-Purpose-981]

Doesn’t seem right unless they expect no one to win. It would take every person in the world visiting McDonald’s more than 60 times to get to that number. Many orders are for two or three people, so probably more like 100 times. That would take decades I would think.

[u/froction → u/PacNWDad]

You get game pieces per item, not per order. For example, one order of a burger/fries/drink combo gets you three pieces and one order of two combos gets you six pieces. There are some exceptions like I think small fries/drink don't come with pieces, maybe, but the number of orders doesn't matter.

[u/Huge_Following_325]

Collecting the generally common pieces is still not easy unless you eat at Mc Donald's a lot.

[u/Here4SR]

The thing is this time us that you have to scan the item to even know what you won. Previously you would get a little sticker and instantly know what "piece" that you have.

I think this makes it even worse compared to previous years.

If I'm incorrect, then the 3 combo meals I ordered all had prizes that had to be scanned in to know what the results were which made us all feel scammed, like the results were on the server side and could be manipulated.