Sunday, February 8, 2015

Roll Up The Win To Win, Odds, Luck, and the Law of Large Numbers

A Warm Cup Full of Hope?

"Roll Up the Rim to Win" season is a yearly warm cup full of hope in the otherwise dreary month of February. There is a car at stake, after all (one for the whole country), and the winner could be me.

Here is how the narrative of luck plays out in our heads:  I realized after the fact that I threw out a cup in Target (speaking of dreary Februarys).  Briefly, it crossed my mind,  "what if it was a big prize?"
We might feel the same way if we threw out a lottery ticket.

Here are some reasons why I shouldn't worry about it:

-The stated odds (from are 1/6 to win any prize. I only had about a 17% chance of winning *any* prize. I can live with forgetting about that cup.

-there are 3 cars to be won in Ontario.  There are some 143,000,000 cups being sold in Ontario.  See here: Restaurant Rules. This is statistically zero. Actually, do some lotteries have better odds?

-there are 94 t.v.s to be won in Ontario.  Again, statistical zero.

-there ARE 23,000 $100 gift cards at stake.  I calculated a 0.16% chance of winning one. Again, I can live with not rolling that cup

-that means the 1/6 odds basically apply to food prizes. There is a 17% chance I missed out on a doughnut.

-the picture above is from the first cup i've had this year.  I didn't roll the second, because I threw it out. Expecting a streak of 5 losses in a row is how you might need to think about it.  Then again, you can sometimes roll a "six" in a dice game, when you need it...

The Law of Large Numbers, and Our Perception of Luck

I will let you read the Wikipedia article on the Law of Large Numbers. The  "house always wins" rule, you might call it.  One of the most brilliant ways to teach it is to use a coin flip simulator, combined with the kids actually flipping their own coins a number of times.  See the Virtual Manipulative here. They watch the sim go to very close to 50/50 over just a few thousand trials, and understand that the odds must come true over time.  How do you think casinos stay in business?  Because they *know* how much they will lose and gain, as an average, on every single bet.

I used to have kids record their flips like this:


There is an amazing trick you can pull on them. Have them record one real trial of 50, and one fake trial.  You will be able to guess which one is fake, every single time!  Why?  Simple-our brains perceive 50/50 as truly random, so kids will make their fake trial *too* random!

Real trials of 50 will usually have a streak like:


Look for those.

How does this apply to Roll Up the Rim?

Let's say you have won 3 doughnuts in a row.  You feel lucky right?  You might even feel like you should go out and buy a lottery ticket.  Wrong- after you have had a streak of good luck is the *worst* time to go out and buy a lottery ticket.

Here is your streak:


But what cups did you have before, and how many after?

Is your true distribution something like this?


4/18, or just ahead of the odds. I suppose that would be a bit lucky.  But there's a reason for the saying "quit while you're ahead..."

We're talking about math here, folks, so this even has a name:  Poisson clumping.  Random events occur in bursts.

Now that we've talked a bit about Roll Up the Rim odds, here are your choices:
-ignore the tabs, because there is a 5/6 chance every single time that you have lost
-keep dreaming of that new car!
-just enjoy your coffee and have fun


  1. I can see this topic and your particular story as an excellent math talk discussion for the class. I had a similar situation when at the airport last year; threw out my cup and felt obligated to go back and search through the trash. My wife convinced me to let it go - not because of the poor expected value, but to save me from embarrassment (and her).

  2. I love your math thinking here, Matthew, but I also love how you made probability a "real world" problem. Yes, this is a game, but it's more than just the game that's played year-to-year in math classes as students explore probability. The Roll Up The Rim Game helps students think about probability in the real world (just like you did here). It reminds me of the Math Task that I created last year after a discussion with my previous VP, Kristi, about probability: Some of my students even chose to explore Roll Up The Rim To Win, and they actually came to some similar conclusions. They told me to just enjoy my coffee, and if I was lucky, maybe I'd win another one (or a donut). I wasn't that lucky! :)