Despite all the advertisements telling us about the lavish lifestyle that comes to the winners of the lottery, most of us know that buying lottery tickets is just a less efficient means of throwing away money. The odds of actually winning a pretty low, as we have pointed out in this article. There was one couple, however, who spotted a loophole in the lotteries of a couple of states. Instead of beating the odds, they used basic math and counted on the odds to rake in millions.

Jerry and Marge Selbee were enjoying their retirement in Evart, Michigan when Jerry noticed something about the lottery WINFall. WINFall allowed customers to purchase a ticket for $2. With each ticket, the customer chose six numbers between 1 and 46. The grand prize is awarded to anyone who happens to pick all six of the numbers that end up being drawn. There are lesser prizes for those who match smaller combinations, such as 5 out of 6 or even 2 out of 6.

What caught Jerry’s attention was a peculiar feature in WINFall. Unlike other lottery games where the jackpot keeps growing until someone wins in the big prize, once the jackpot reached $5 million without a winner, the prize amount would roll down to those who had tickets that matched 5, 4, and 3 out of the six numbers.

Jerry said he always had “a head for math,” and he holds a degree in mathematics from Western Michigan University. When he heard about the roll-down feature of WINFall, he sat down and started crunching some numbers.

What follows is a grossly-simplified explanation of how he ended up making a fortune. If you are interested in delving in much greater detail into the statistical analysis, feel free to read this article from the Journal of American Science.

To understand his strategy, we need to grasp the concept of “expected value.” Imagine that the grand prize of a lottery game is $2. The price of a lottery ticket is $1. Rather than picking numbers, you pick either heads or tails of a coin. Instead of pulling numbers out of a bowl, the lottery director simply flips a coin. If you successfully guess the result, you win the $2. If you are wrong, you get nothing and lose the $1 that you spent on your ticket.

In this scenario, you can expect to guess correctly half of the time, since there are only two possibilities in a coin toss. The odds of guessing correctly, therefore, are 1 out of 2, or 50%. Since the grand prize is $2, you can calculate the expected value by multiplying 50% by $2, giving us $1. In other words, on average, your $2 ticket will win you $1.

It doesn’t take a mathematical genius to figure out that the coin-flip lottery is not the best of investment strategies. With an expected value of $1, anyone who sinks $1,000 in lottery tickets can expect to walk away with $500.

Now let’s look at the game that Jerry and Marge Selbee played. For purposes of this analysis, we’re going to look at the actual numbers from the February 8, 2010, Massachusetts WINFall jackpot. (Thanks to Half as Interesting for explaining the math in this video.)

Assume we purchase 200,000 tickets at $2 each. Our initial investment is $400,000.

The odds of matching 5 out of 6 numbers is 1 in 39,028. That means approximately 5.12 of our tickets will be winners. The prize for matching 5 out of 6 is normally $4,000, but because of the jackpot roll-down, it increased to $22,096. If we multiply our 5.12 winning tickets by $22,096, we get the expected value of $110,480.

Since we started with $400,000, winning a paltry $110,480 doesn’t seem like much of a deal, does it? Hold on, though. WINFall also awards prizes for smaller combinations.

A prize of $807.52 was awarded for those who matched 4 out of 6 numbers. The odds of matching 4 out of 6 numbers is 1 in 800. That means about 250 of our tickets will be winners. Multiply $807.52 by 250, and we have an expected value of $201,880. Add that to the expected value from the 5 out of 6 combinations, and we have a total expected value of $312,360.

Those who matched 3 out of 6 qualified for a payout of $26.85. The odds of making this match are 1 out of 47.4. This means 4,219 of our tickets should be winners. At $26.85 apiece, those tickets would generate an expected value of $201,880. Add that to the previous expected values, and we are at $425,640. This means our initial $400,000 investment is starting to pay off.

There are even benefits if you match just 2 out of the 6 numbers. Those who do so qualify for a free lottery ticket that ordinarily costs $2. Odds of making this match are 1 in 6.8. This means 29,412 of our tickets should qualify, earning us an expected value of $58,823. This brings our total up to $484,206 — a return of more than 20% from our initial investment.

But wait… We haven’t addressed the odds of winning the grand prize. The likelihood of matching all six numbers is 1 in 9,366,819. Add this into the equation, and our grand total expected value from the $400,000 lottery ticket purchase is $547,881.

Once Jerry checked and double-checked his math, he talked to his wife about supplementing their retirement savings. On their first attempt, they purchase $3,600 worth of WINFall tickets. That earned them $6,300. The next time, they sank $8,000 into tickets. They doubled that amount, as well. It wasn’t long before they were buying hundreds of thousands of dollars worth of tickets in WINFall.

They were so successful in this venture that they established a company, G.S. Investment Strategies, and sold interest of their business to family and friends for $500 per share. By the time Michigan discontinued WINFall in 2005, G.S. Investment Strategies had 25 members.

Although Michigan’s WINFall game was drawing to a close, the Selbees learned that Massachusetts had started a similar program called “Cash WinFall.”

For the next six years, the Selbees drove 14-hours to Massachusetts whenever the grand prize reached the point where there would be a roll-down. They bought hundreds of thousands of tickets at two convenience stores and then spent 10 hours a day at a nearby motel room to sort through the tickets.

In 2011, the Boston Globe ran a story about massive Cash WINFall ticket sales from a few locations. They discovered the Selbees and a group of MIT math students were behind almost all of the sales. The newspaper story triggered an investigation by the state’s lottery commission. The investigation concluded that everything the Selbees and others had been doing was perfectly legal.

The lottery commission decided to terminate the Cash WINFall game, but by this point, the Selbees had won a total of $26 million. That doesn’t count the money they made from selling the rights to turn their story into a motion picture.

Read more fun facts about mathematics.

Read more fun facts about money.