Thursday, October 09, 2014

The Mathematics of McDonald's Monopoly (2014)

I designed and now teach a course at my college called The Mathematics of Games and Gambling.  In that course, we look at things like probability, odds, and expected values.  In addition, since the course is aimed at non-mathematics majors, I try to work in current events (such as large lotteries or March Madness during spring semesters).  Since I'm teaching the course in the fall this year, I didn't have March Madness to use as my current event - and so I went with a much less healthy option:  McDonald's Monopoly promotion.

When I first decided to use McDonald's Monopoly as a lesson, I immediately went to the internet to see if someone else had already gone through the trouble of figuring out the various probabilities for me.  At first, I found a promising article but it ended up being a direct copy of last year's rules and numbers which didn't help me for this year at all.

Then, I stumbled across this blog entry written by Michael Ross which deals with the 2014 numbers and has now spread all across the web.

I was, of course, elated at first to find that someone else had done all the work for me.  But as any good instructor, I decided I had better first verify the values that Mr. Ross came up.  Unfortunately, the first value that I tried to verify was the number of plays one would need to have at least a 50% chance at landing Mediterranean Avenue.  It turned out that I was unable to reproduce Mr. Ross' numbers.  Oh sure, my values are generally close to his, but not exact.

As such, with the help of a colleague and Mathematica, I decided to redo all the values and go with my numbers - so here are my results (I used Mathematica to crunch the values for me - I am guessing Mr. Ross used something like Excel which produced some rounding errors?).  I suppose I could have always screwed up on my end too - I'm not above saying that could be the case rather than Mr. Ross making the error.

Anyhow, since Mr. Ross didn't provide any mathematical work for his values (which makes double-checking his work impossible), I thought I would provide my work for others to check if they would like:


Here NumberOfPlays is a function that takes the given probability of a particular prize and it returns the number of plays you would need to have at least a 50% chance of winning said prize.  As I said, this was all done in Mathematica.

Mediterranean Avenue:

You need 451,822 plays in order to have at least a 50% shot at landing the rare brown property.  That would mean spending over $450,000 on hashbrowns (the cheapest food item containing game pieces) in order to win the $50 prize.  Ouch.

Here's the work in Mathematica using the function defined above.


Vermont Avenue:

You would need 112,955,545 plays in order to land the rare light blue property.  If you were to buy medium fountain drinks at McDonald's, you'd fill over 28 Olympic sized swimming pools with soda before having at least a 50% chance of landing the rare piece.

Virginia Avenue:

For some reason, Mr. Ross's value matched mine here (the only value that matched by the way).  You need 90,364,428 plays to have a 50% shot at the property.  My college has 1,357 students enrolled this year which means if you bought 10-piece nuggets you'd have 665,913 nuggets per student at my school by the time you'd have a 50% chance of landing Virginia Avenue.  College students do like their chicken nuggets but I think 665,000+ nuggets per person is a bit excessive.

Tennessee Avenue:

You need 1,898,412 plays to have a 50% chance of landing Tennessee Avenue.  That's a lot of junk food for the hopes of getting a new cell phone.

Kentucky Avenue:

You need 22,591,109 plays to have a 50% chance of getting Kentucky Avenue.  If you were to buy Big Macs for each of your plays and start stacking them to the top of the Empire State Building (the tip of the antenna), you'd end up with 3,560 stacks of Big Macs reaching the top of the building (and still have another stack that reaches over half the height of the building).

Ventnor Avenue:

You need 6,024,295 plays to have a 50% chance of landing Ventnor Avenue (and thus winning a Beaches Resort Vacation).  If you tried to win via buying only Bacon Clubhouse Burgers (at $4.69 a piece), you could instead buy 22,177 Royal Caribbean 7-day cruises (at $1,274 per cruise) and still have a bit of cash left over.

Pennsylvania Avenue:

You need 225,911,097 plays to have a 50% chance of landing Pennsylvania Avenue.  That's 37,651,849.5 gallons of large Hot McCafe drinks if you tried to play that way.




Boardwalk:

You need 451,822,161 plays to have a 50% chance at finding Boardwalk (and the $1,000,000 prize that goes with it).  If you tried to find Boardwalk via Big Macs and you placed each Big Mac side-by-side along the equator, the string of burgers would stretch all the way around the world and then a bit more (1.0739 times around the equator to be precise)!

As you can see, the odds of landing any of the "big" prizes is quite small...but if you like buying McDonald's food anyhow, the promotion is a nice little bonus (and you are likely to win something - 1 in 4 game pieces is a winner of some sort).
I'm guessing the codes are all used.

PS:  Did you know?  The way to remember the list of rare pieces in McDonald's Monopoly is that with the exception of Boardwalk (the rarest piece), each color set always has the property that comes last alphabetically as the rare piece.  For example, the yellow properties of Atlantic Ave, Ventnor Ave, and Marvin Gardens.  Since Ventnor is last alphabetically, that is the rare yellow property!

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