Winning the Lottery An Expected Value Mystery


by Michelle Manes M


any of our sessions at the Math Teachers’ Circle of Hawai’i (MaTCH) are focused on giving teachers time to explore and problem solve, and the facilitators don’t


spend too much time “teaching” or explaining any particular mathematical content. Sometimes, though, we find an irresistible piece of mathematics that we want to share at MaTCH, and these sessions run a bit differently:


• Introductory activity to inspire / elicit a particular mathematical idea


• Teaching about that idea (usually a 10-15 minute PowerPoint or whiteboard talk followed by Q&A time)


• Follow-up activity that either extends the math- ematical idea or ties it to other topics


The Inspiration


Here I’ll describe a recent MaTCH session with this format, inspired by my friend Jordan Ellenberg’s wonderful book How Not to Be Wrong: The Power of Mathematical Thinking. Ellenberg tells the story of a lottery scam that wasn't actually a scam:


In 2005, an MIT student was working on a senior


project examining the expected value for lottery tickets in various states, and he noticed something incredible. On certain special weeks, the expected value of a $2 Cash Winfall ticket in Massachusetts was about $5.53. (In most weeks, the expected value for a ticket in


this lottery, like in all lotteries, was much lower than the ticket price: 80¢ for a $2.00 ticket. So those special weeks were really, really special.) Of course, that doesn’t mean that every $2 ticket


would pay out $5.53. As Ellenberg says, “expected value is not the value you expect.” What it does mean is that if you buy lots and lots of tickets — thousands of them — then you have a good chance of almost tripling your money.


⊆ 08 Aſter checking and re-checking and re-re-checking


his work, this student got together with several of his friends and did just that. Tey bought thousands of lottery tickets each time the lottery had this special, high expected value week. And they won millions of dollars between 2005 and the last Cash Winfall drawing in 2012. Two other “cartels” noticed the same opportunity


in the Massachusetts lottery and also started playing big on those special weeks. But the MIT cartel was different: Instead of using “Quick Pick” machines to choose numbers randomly, the MIT students under- took the tedious job of filling out their lottery tickets by hand.


Ellenberg asks the obvious question: Why would


anyone do that? He offers two explanations: First, birthday problem-like reasoning says that using the Quick Pick machine thousands of times will surely result in lots of duplicate tickets, something the cartels want to avoid. His second explanation has to do with guaranteeing a minimum return on the bet, and it ties to even more interesting mathematics: finite projective geometries.


Introductory Activity We started the session by playing a mini lottery game. Each table had a banker to collect money for ticket purchases before each round and to pay out winnings aſter each draw. Each player started with $10 in Mo- nopoly money. Te rules:


• We used standard playing cards numbered 1–7. A “ticket” was a choice of three numbers from 1–7, no repeats.


• Before each round, participants bought their tickets for $1 each. Tey decided which tickets to buy and how many.


• Each round, the MC shuffled the mini-deck, drew 3 cards, and placed them face-up on the document camera.


MTCircular · Summer/Autumn 2016 · American Institute of Mathematics


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