When Lawrence Mower, an investigative reporter in Florida, examined a list of over a million prizes worth \$600+ that had been claimed in the Florida lottery, he noticed something interesting—there were a few outliers in the data. Not a couple of people with somewhat high winnings, either: there were several people who had won more than 100 of these prizes. One man from Pompano Beach, Florida had 252 wins worth \$600 or more; another Hollywood, FL resident had claimed 578.
What was suspicious about these numbers? Garibaldi mentioned a few possibilities. Clerks verify that a ticket is a winner by scanning a barcode. If a gambler approaches a clerk with a winning ticket, the clerk can claim it’s a loser or worth less than it actually is, and keep the winnings for themselves. If a ticket is worth a lot of money—over \$600 in Florida—then gamblers need to travel to the state lottery office and register themselves or their earnings for tax purposes. If a gambler wants money right away or is wary of registering themselves with the state government, they might be willing to sell their tickets for less than they’re worth, letting the buyer claim the winnings. These buyers are called ticket aggregators, and in some cases they even use this system for money laundering.
When Mower asked the Florida lottery secretary about his findings, she dismissed them. But Mower wasn’t convinced that he hadn’t found evidence of an illicit scheme, so he contacted a few mathematicians to work out how likely it really was that someone could win so many huge lottery prizes. Among those mathematicians was Skip Garibaldi, who related all of this at the start of his invited address this morning, “Uncovering lottery shenanigans.” Garibaldi talked about he, along with Philip Stark of UC Berkeley and Richard Arratia of USC, worked out how much a gambler would need to spend in order to have a non-negligible probability of the kinds of winnings that Mower had discovered. You can probably guess how it turned out, but you might be impressed by how extreme the numbers really are—I certainly was.
First, they needed to decide what a “non-negligible probability” meant. There were a few options, but they went with their most conservative option of a one in a million chance. If a gambler is playing several different games, you can construct a vector of ticket costs and a vector of number of tickets bought at each game. In the simplest case, the games are all scratch-off tickets, meaning the probability of winning is described by a binomial distribution. By minimizing over the number of tickets bought under the condition that the probability of getting the desired number of wins is nonnegligible, the team found an estimate of how much a gambler with hundreds of wins should be spending.
What they found was that our Pompano Beach friend should have been spending roughly \$1,000 per day on lottery tickets. That’s a lot of lottery tickets—and an interview with Mower revealed that he was, in fact, a ticket aggregator. (The Hollywood gambler was innocent.) The results have prompted policy changes across the US to address the issues of ticket aggregation and money laundering.