One window depicts a spiraling DNA helix, a nod to former college fellow Francis Crick. Another shows a trio of overlapping circles in tribute to John Venn. There is also a checkerboard situated in the glass, each square colored in a seemingly random way. It’s there to commemorate one of the founders of modern statistics, Ronald Fisher.
After winning a scholarship at Gonville and Caius, Fisher spent three years studying at Cambridge, specializing in evolutionary biology. He graduated on the eve of the First World War and tried to join the British Army. Although he completed the medical exams several times, he failed on each occasion because of poor eyesight. As a result, he spent the war teaching mathematics at a number of prominent English private schools, publishing a handful of academic papers in his spare time.
As the conflict drew to a close, Fisher began to search for a new job. One option was to join Karl Pearson’s laboratory, where he had been offered the role of chief statistician. Fisher wasn’t particularly keen on this option: the previous year, Pearson had published an article criticizing some of his research. Still reeling from the attack, Fisher declined the job.
Instead, Fisher took a job at the Rothamsted Experimental Station, where he turned his attention to agricultural research. Rather than just being interested in the results of experiments, Fisher wanted to make sure that experiments were designed to be as useful as possible. “To consult the statistician after an experiment is finished is often merely to ask him to conduct a post mortem examination,” he said. “He can perhaps say what the experiment died of.”
Considering the work at hand, Fisher was puzzled about how to scatter different crop treatments across a plot of land during an experiment. The same problem appears when conducting medical trials across a large geographic area. If we are comparing several different treatments, we want to make sure they are scattered across a wide region. But if we distribute them by picking locations at random, there is a chance that we will repeatedly pick similar locations. In which case, a treatment ends up concentrated only in one area, and we have a pretty lousy experiment.
Suppose we want to test four treatments across sixteen trial sites, arranged in a four-by-four grid. How can we scatter the treatments across the area without risking all of them ending up in the same place? In his landmark book The Design of Experiments, Fisher suggested that the four treatments be distributed so that they appear in each row and column only once. If the field had good soil at one end and poor land at the other, all treatments therefore would be exposed to both conditions. As it happened, the pattern Fisher proposed had already found popularity elsewhere. It was common in classical architecture, where it was known as a Latin square, as shown in Figure 2.1.
FIGURE 2.1.
The stained glass window at Gonville and Caius College shows a larger version of a Latin square, with the letters—one for each type of treatment—replaced by colors. As well as earning a tribute in an ancient hall, Fisher’s ideas are still used today. The problem of how to construct something that is both random and balanced arises in many industries, including agriculture and medicine. It also comes up in lottery games.
Lotteries are designed to cost players money. They originated as a palatable form of tax, often to support major building projects. The Great Wall of China was financed with profits from a lottery run by the Han dynasty; proceeds from a lottery organized in 1753 funded the British Museum; and many of the Ivy League universities were built on takings from lotteries arranged by colonial governments.
Modern lotteries are made up of several different games, with scratchcards a lucrative part of the business. In the United Kingdom, they make up a quarter of the National Lottery’s revenues, and American state lotteries earn tens of billions of dollars from ticket sales. Prizes run into the millions, so lottery operators are careful to limit the supply of winning cards. They can’t put random numbers below the scratch-off foil, because there is a chance that could produce more prizes than they could afford to pay out. Nor would it be wise to send batches of cards to places arbitrarily, because one town could end up with all the “lucky” tickets. Scratchcards need to include an element of chance to make sure the game is fair, but operators also need to tweak the game somehow to ensure that there aren’t a huge number of winners or too many in one place. To quote statistician William Gossett, they need “controlled randomness.”
FOR MOHAN SRIVASTAVA, THE idea that scratchcards follow certain rules started with a joke present. It was June 2003, and he’d been given a handful of cards, including one with a collection of tic-tac-toe games. When he scratched off the foil, he discovered three symbols in a line, which netted him three dollars. It also got him thinking about how the lottery keeps track of the different prizes.
Srivastava worked as a statistician in Toronto, and he suspected that each card contained a code that identified whether it was a winner. Code breaking was something he had always found interesting; he’d known Bill Tutte, the British mathematician who had broken the Nazi Lorenz cipher in 1942, an achievement later described as “one of the greatest intellectual feats of World War II.” On the way to collect his prize from a local gas station, Srivastava started to wonder how the lottery might go about distributing the tic-tac-toe scratchcards. He had plenty of experience with such algorithms. He worked as a consultant for mining companies, which hired him to hunt down gold deposits. In high school, he’d even written a computer version of tic-tac-toe for an assignment. He noticed that each foil panel on the scratchcard had a three-by-three grid of numbers printed on it. Perhaps these numbers were the key?
Later that day, Srivastava stopped by the gas station again and bought a bundle of scratchcards. Examining the numbers, he found that some numbers appeared several times on the card, and some only once. As he sifted through a pile of cards, he spotted the fact that if a row contained three of these unique numbers, it usually signaled a winning card. It was a simple and effective method. The challenge was finding such cards.
Unfortunately, it turns out that winning cards aren’t all that common. For example, during the early hours of April 16, 2013, a car smashed through the doors of a convenience store in Kentucky. A woman jumped out, grabbed a display containing 1,500 scratchcards, and drove away. By the time she was arrested a few weeks later, she’d managed to claim a mere $200 in prizes.
Even though Srivastava had a reliable—and legal—method for finding profitable scratchcards, it didn’t mean he could turn it into a lucrative business. He worked out how long it would take to sort through all potential cards to find the “lucky” ones and realized that he was better off sticking with his existing job. Having decided it wouldn’t be worth changing careers, Srivastava thought the lottery might like to know about his discovery. First he tried to get ahold of them by phone, but, perhaps thinking he was just another gambler with a dodgy system, they didn’t return his calls. So, he divided twenty untouched scratchcards into two groups—one of winners, and one of losers—and mailed them to the lottery’s security team by courier. Srivastava got a phone call from the lottery later that day. “We need to talk,” they said.
The tic-tac-toe games were soon removed from stores. According to the lottery, the problem was due to a design flaw. But since 2003, Srivastava has looked at other lotteries in the United States and Canada and suspects some may still be producing scratchcards with the same problem.
In 2011, a few months after Wired magazine featured Srivastava’s story, reports emerged of an unusually successful scratchcard player in Texas. Joan Ginther had won four jackpots in the Texas scratchcard lottery between 1993 and 2010, bringing in a total of $20.4 million. Was it just down to luck? Although Ginther has never commented on the reason for her multiple wins, some have speculated that her statistics PhD might have had something to do with it.
It’s not just scratchcards that are vulnerable to scientific thinking. Traditional lotteries do not include controlled randomness, yet they are still not safe from mathematically inclined players. And when lott
eries have a loophole, a winning strategy can begin with something as innocuous as a college project.
EVEN WITHIN A UNIVERSITY as famously offbeat as the Massachusetts Institute of Technology, Random Hall has a reputation for being a little quirky. According to campus legend, the students who first lived there in 1968 wanted to call the dorm “Random House” until the publisher of that name sent them a letter to object. The individual floors have names, too. One is called Destiny, a result of its cash-strapped inhabitants selling the naming rights on eBay; the winning bid was $36 from a man who wanted to name it after his daughter. The hall even has its own student-built website, which allows occupants to check whether the bathrooms or washing machines are available.
In 2005, another plan started to take shape in the corridors of Random Hall. James Harvey was nearing the end of his mathematics degree and needed a project for his final semester. While searching for a topic, he became interested in lotteries.
The Massachusetts State Lottery was set up in 1971 as a way of raising extra revenue for the government. The lottery runs several different games, but the most popular are Powerball and Mega-Millions. Harvey decided that a comparison of the two games could make for a good project. However, the project grew—as projects often do—and Harvey soon began to compare his results with other games, including one called Cash WinFall.
The Massachusetts Lottery introduced Cash WinFall in autumn 2004. Unlike games such as Powerball, which were played in other states too, Cash WinFall was unique to Massachusetts. The rules were simple. Players would choose six numbers for each two-dollar ticket. If they matched all six in the draw, they won a jackpot of at least half a million dollars. If they matched some but not all the numbers, they won a smaller sum. The lottery designed the game so that $1.20 of every $2.00 would be paid out in prizes, with the rest being spent on local good causes. In many ways, WinFall was like all the other lottery games. However, it had one important difference. Usually, when nobody wins the jackpot in a lottery, the prize rolls over to the next draw. If there’s no winning ticket next time, it rolls over again and continues to do so until somebody eventually matches all the numbers. The problem with rollovers is that winners—who are good publicity for a lottery—can be rare. And if no smiling faces and giant checks appear in the newspapers for a while, people will stop playing.
Massachusetts Lottery faced precisely that difficulty in 2003, when its Mass Millions game went without a winner for an entire year. They decided that WinFall would avoid this awkward situation by limiting the jackpot. If the prize money rose to $2 million without a winner, the jackpot would “roll down” and instead be split among the players who had matched three, four, or five numbers.
Before each draw, the lottery published its estimate for the jackpot, which was based on ticket sales from previous draws. When the estimated jackpot reached $2 million, players knew that the money would roll down if nobody matched six numbers. People soon spotted that the odds of winning money were far better in a roll-down week than at other times, which meant ticket sales always surged before these draws.
As he studied the game, Harvey realized that it was easier to make money on WinFall than on other lotteries. In fact, the expected payoff was sometimes positive: when a roll down happened, there was at least $2.30 waiting in prize money for every $2.00 ticket sold.
In February 2005, Harvey formed a betting group with some of his fellow MIT students. About fifty people chipped in for the first batch of tickets—raising $1,000 in total—and tripled their money when their numbers came up. Over the next few years, playing the lottery became a full-time job for Harvey. By 2010, he and a fellow team member incorporated the business. They named it Random Strategies Investments, LLC, after their old MIT accommodation.
Other syndicates got in on the action, too. One team consisted of biomedical researchers from Boston University. Another group was led by retired shop owner (and mathematics graduate) Gerald Selbee, who had previously had success with a similar game elsewhere. In 2003, Selbee had noticed a loophole in a Michigan lottery game that also included roll downs. Gathering a thirty-two-person-strong betting group, Selbee spent two years bulk-buying tickets—and netting jackpots—before the lottery was discontinued in 2005. When Selbee’s syndicate heard about WinFall, they turned their attention to Massachusetts. There was a good reason for the influx of such betting teams. Cash WinFall had become the most profitable lottery in the United States.
DURING THE SUMMER OF 2010, the WinFall jackpot again approached the roll-down limit. After a $1.59 million prize went unclaimed on August 12, the lottery estimated that the jackpot for the next draw would be around $1.68 million. With a roll down surely only two or three draws away, betting syndicates started to prepare. By the end of the month, they planned to have thousands more dollars in winnings.
But the roll down didn’t arrive two draws, or even three draws, later. It came the following week, on August 16. For some reason, there had been a huge increase in ticket sales, enough to drive the total prize money past $2 million. This flood of sales triggered a premature roll down. The lottery officials were as surprised as anyone: they had never sold that many tickets when the estimated jackpot was so low. What was going on?
When WinFall was introduced, the lottery had looked into the possibility of somebody deliberately nudging the draw into a roll down by buying up a large number of tickets. Aware that ticket sales depended on the estimated jackpot—and potential roll downs—the lottery didn’t want to get caught out by underestimating the prize money.
They calculated that a player who used stores’ automated lottery machines, which churned out tickets with arbitrary numbers, would be able to place one hundred bets per minute. If the jackpot stood at less than $1.7 million, the player would need to buy over five hundred thousand tickets to push it above the $2 million limit. Because this would take well over eighty hours, the lottery didn’t think anyone would be able to tip the total over $2 million unless the jackpot was already above $1.7 million.
The MIT group thought otherwise. When James Harvey first started looking at the lottery in 2005, he’d made a trip to the town of Braintree, where the lottery offices were based. He wanted to get ahold of a copy of the guidelines for the game, which would outline precisely how the prize money was distributed. At the time, nobody could help him. But in 2008, the lottery finally sent him the guidelines. The information was a boost for the MIT group, which until then had been relying on their own calculations.
Looking at past draws, the group found that if the jackpot failed to top $1.6 million, the estimate for the next prize was almost always below the crucial value of $2 million. Pushing the draw over the limit on August 16 had been the result of extensive planning. As well as waiting for an appropriate jackpot size—one close to but below $1.6 million—the MIT group had to fill out around 700,000 betting slips, all by hand. “It took us about a year to ramp up to it,” Harvey later said. The effort paid off: it’s been estimated that they made around $700,000 that week.
Unfortunately, the profits did not continue for much longer. Within a year, the Boston Globe had published a story about the loophole in WinFall and the betting syndicates that had profited from it. In the summer of 2011, Gregory Sullivan, Massachusetts Inspector General, compiled a detailed report on the matter. Sullivan pointed out that the actions of the MIT group and others were entirely legal, and he concluded that “no one’s odds of having a winning ticket were affected by high-volume betting.” Still, it was clear that some people were making a lot of money from WinFall, and the game was gradually phased out.
Even if WinFall hadn’t been canceled, the Boston University syndicate told the inspector general that the game wouldn’t have remained profitable for betting teams. More people were buying tickets in roll-down weeks, so the prizes were split into smaller and smaller chunks. As the risk of losing money increased, the potential rewards were shrinking. In such a competitive environment, it was crucial to obtain an edge over other team
s. The MIT group did this by understanding the game better than many of their competitors: they knew the probabilities and the payoffs and exactly how much advantage they held.
Betting success is not just limited by competition, however. There is also the not-so-small matter of logistics. Gerard Selbee pointed out that if a group wanted to maximize their profits during a roll-down week, they needed to buy 312,000 betting slips, because this was the “statistical sweet spot.” The process of buying so many tickets was not always straightforward. The ticket machines would jam in humid weather and run slowly when low on ink. On one occasion, a power outage got in the way of the MIT group’s preparations. And some stores would refuse to serve teams altogether.
There was also the question of how to store and organize all the tickets they bought. Syndicates had to keep millions of losing tickets in boxes to show to tax auditors. Moreover, it was a headache to find the winning slips. Selbee claims to have won around $8 million since he starting tackling lotteries in 2003. But after a draw, he and his wife would have to work for ten hours a day examining their collection of tickets to identify the profitable ones.
SYNDICATES HAVE LONG USED the tactic of buying up large combinations of numbers—a method known as a “brute force attack”—to beat lotteries. One of the best-known examples is the story of Stefan Klincewicz, an accountant who hatched a plan to win the Irish National Lottery in 1990. Klincewicz had noticed that it would cost him just under £1 million to buy enough tickets to cover every potential combination, thereby guaranteeing a winning ticket when the draw was made. But the strategy would only work if the jackpot was big enough. While waiting for a large rollover to appear, Klincewicz gathered a twenty-eight-man syndicate. Over a period of six months, the group filled out thousands upon thousands of lottery tickets. When a rollover of £1.7 million was eventually announced for the bank holiday draw in May 1992, they put their plan into action. Picking lottery terminals in quieter locations, the team started placing the necessary bets.
The Perfect Bet Page 4