Statistician cracks Toronto lottery

Christian points me to this amusing story by Jonah Lehrer about Mohan Srivastava, (perhaps the same person as R. Mohan Srivastava, coauthor of a book called Applied Geostatistics) who discovered a flaw in a scratch-off game in which he could figure out which tickets were likely to win based on partial information visible on the ticket. It appears that scratch-off lotteries elsewhere have similar flaws in their design.

The obvious question is, why doesn’t the lottery create the patterns on the tickets (including which “teaser” numbers to reveal) completely at random? It shouldn’t be hard to design this so that zero information is supplied from the outside. in which case Srivastava’s trick would be impossible.

So why not put down the numbers randomly? Lehrer quotes Srivastava as saying:

The tickets are clearly mass-produced, which means there must be some computer program that lays down the numbers. Of course, it would be really nice if the computer could just spit out random digits. But that’s not possible, since the lottery corporation needs to control the number of winning tickets. The game can’t be truly random. Instead, it has to generate the illusion of randomness while actually being carefully determined.

I’d phrase this slightly differently. We’re talking about $3 payoffs here, so, no, the corporation does not need to control the number of winning tickets. What they do need to control is the probability of a win, but that can be done using a completely random algorithm.

From reading the article, I think the real reason the winning tickets could be predicted is that the lottery tickets were designed to be misleadingly appealing. Lehrer writes:

Instead of just scratching off the latex and immediately discovering a loser, players have to spend time matching up the revealed numbers with the boards. Ticket designers fill the cards with near-misses (two-in-a-row matchups instead of the necessary three) and players spend tantalizing seconds looking for their win. No wonder players get hooked.

“Ticket designers fill the cards with near-misses . . .”: This doesn’t sound like they’re just slapping down random numbers. Instead, the system seems to be rigged in the fashion of old-time carnival games in order to manipulate one’s intuition that the probability of near-misses should be informative about the underlying probability of hits. (See here for some general discussion of the use of precursors to estimate the probability of extremely rare events.)

In this sense, the story is slightly more interesting than “Lottery designers made a mistake.” The mistake they made is directly connected to the manipulations they make in order to sucker people into spend more money.

P.S. Lehrer writes that Srivastava does consulting. This news story should get him all the business he needs for awhile!

5 thoughts on “Statistician cracks Toronto lottery

  1. I thought about this too, but random assignment by itself would not do the trick for this type of scratcher. For one thing, it would make big wins equally likely as small ones, and, if I understand the game, it would increase the chances of getting multiple wins on the same card.

    Even if you adjusted the payments to keep the expected value positive for the issuer, this is still not the reinforcement scheme you'd want. Lots of small payouts and the hope of a big one are what keep the people coming back.

    (this is, of course, the most regressive way of raising money I can think of but that's a different topic)

  2. It looks like they give you 8 tic-tac-toe boards with numbers 1-39, then you get 24 numbers from your scratch off, leaving only 15 you don't have. It takes at least 3 numbers missing to prevent any tic-tac-toe, 5 numbers if they allow you to get the center square. So they need to fill somewhere between 24 and 40 squares (depending on how many centers you get) to prevent any tic-tac-toes. This means reusing the 15 numbers you don't have and so the numbers that block you tend to be used multiple times. So he looks for singletons (numbers which only appear once) and if they win then likely so does the card.

    That said, this could be easily fixed. Hold back one killer number. Design a winning card. Print it. Kill the win with your held back number, print the losers. (You could even scramble the actual numbers being used and do various transforms on the losing cards to hide this.) But anyway no info till you scratch off the latex.

  3. How ever it's done, the winner is the lottery's supporting agency. I think the lottery is a tax on the weak minded. State's that capatailize on lotteries are doing a disservice to their public.

  4. I see an interesting back-story here.

    In fact, it reminded him a lot of his day job, which involves consulting for mining and oil companies. … "I remember thinking, I’m gonna be rich! I’m gonna plunder the lottery!" … "I estimated that I could expect to make about $600 a day. That’s not bad. But to be honest, I make more as a consultant, and I find consulting to be a lot more interesting than scratch lottery tickets."

    I wonder if Mr. Srivastava ever pondered the probability and statistics of finite resources and perhaps alerting the media on the rigged nature of unlimited plundering?

  5. I have to agree with Basil here, a win in a lottery is just a bait for weak-minded individuals. They get a bite and they don't stop chewing. Problem is, for every chow they have to pay more and more and in the end, lottery is always the winner.


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