When Jordan Ellenberg spoke at the MAA Carriage House on June 11, he regaled the crowd with a story from his book How Not to Be Wrong: The Power of Mathematical Thinking, new from Penguin Press.
In “How to Get Rich Playing the Lottery,” the University of Wisconsin–Madison mathematician told a ripped-from-the-headlines tale that typifies, he said, “the way that rather simple mathematical ideas can actually branch out and have rather broad applications to a surprising set of things.”
To even begin to think sensibly about lotteries you have to understand expected value. Suppose a lottery ticket costs $2 and carries a one in 200 chance of winning $300. The expected value of the ticket is $1.50 (because 1/200*300+199/200*0=1.5). But what does this figure mean exactly? It certainly isn’t the value you expect your ticket to have, since the ticket is worth either zero dollars or $300. No other value is possible.
“If we could go back to the dawn of mathematical nomenclature and change things around, we would probably call this something more like ‘average value,’” Ellenberg said. “It’s supposed to represent the average value of lots of randomly selected tickets. It gives you a good clue as to what’s going to happen if you—as most lottery players do—play this game for a long time, which is that you’re going to lose.”
The story of Cash WinFall, though, is the story of long-time lottery players who did not lose—spectacularly so. And expected value can help explain why.
On February 7, 2005 the expected value of a $2 Cash WinFall ticket was a whopping $5.53. In an effort to encourage more people to play the lottery, Massachusetts had adopted a rolldown rule. Whenever the jackpot exceeded $2 million, if no one hit all six numbers, the money in the jackpot was distributed among the winners of the lesser prizes, those whose tickets matched five or four or three of the numbers drawn.
The rolldown rule, Ellenberg said, “actually created a game that was a good idea to play.”
This did not escape notice.
Before the final Cash WinFall drawing in January 2012, three betting syndicates—a group of MIT undergraduates, a family from Michigan, and a biomedical researcher from Northeastern University and his cronies—had exploited the rolldown rule to rake in profits. They bought hundreds of thousands of tickets on rolldown days, consistently making a 10-15% return on investment.
So how did they get away with this? How did Massachusetts not know what was happening?
Massachusetts did know, Ellenberg argued, and actually benefited from the actions of the betting syndicates. The state’s take from the lottery is $0.80 per ticket sold, he explained, which means that the state cares only about how many tickets are sold, not about who wins.
“So if these guys come and were like ‘We’re going to make your lottery a lot bigger by buying hundreds of thousands of extra tickets,’” Ellenberg explained, “that is great for the state.”
$120 million dollars of extra revenue great, in fact.
The money the syndicates made, Ellenberg clarified, came from the other lottery players, those who bought tickets even when the jackpot was not going to trickle down to swell the smaller prizes.
While two of the three syndicates bought their thousands of tickets at random using a Quick Pick machine, the MIT students laboriously bubbled in numbers by hand.
Why would they expend the time and energy to do this when the expected value of a ticket is the same no matter what ticket it is?
“Sometimes in math when you’re trying to explain something the best way to do it is to replace the problem you’re studying with a smaller or easier one which has the same features but somehow you can hold it in your head all at once,” Ellenberg said.
He then used the example of the smaller-scale—three numbers are chosen out of seven possibilities—Transylvanian Lottery to illustrate how studied ticket selection can if not boost the expected value of your winnings, at least reduce their variance.
While in the toy Transylvanian example the Fano plane provides a handy guide to triplet selection, scaling up to the real 46-choose-6 lottery takes some doing. With the help of a 1976 combinatorial design paper by R. H. F. Denniston, though, Ellenberg was able to figure out how the MIT undergrads might have ensured—mathematically—that they didn’t lose the money entrusted to them by their friends and classmates.
“I was able to put together something which I would say, if they weren’t doing this, I think they should have done this,” Ellenberg said. “It’s at least as good as what they did.”
Of course, since no state runs a rolldown lottery anymore, Ellenberg will have to look to book sales for his fortune.—Katharine Merow
Watch a short version of the lecture on YouTube.
Listen to audio of Jordan Ellenberg's lecture (mp3)
Listen to an interview with Jordan Ellenberg and MAA Director of Publications Ivars Peterson (mp3)
This MAA Distinguished Lecture was funded by the National Security Agency.