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A Fun DIY Science Goodie: How to Get a Positive Expected Rate of Return on a Lottery Ticket

The views expressed are those of the author and are not necessarily those of Scientific American.

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So goes popular opinion: the lottery’s an egregious societal evil implemented and overseen by shape-shifting, blood-drinking reptilian aliens. And that may be largely true – designed to slowly and quietly bleed dry your pockets – that is, unless you learn to drive it.

Assuming drawings actually are random, all the science in the world can’t help you pick the winning numbers. But some fiendishly simple stats can make the dollar you put down likely to win back that dollar and more.

For my book, Brain Trust, I interviewed Emory Mathematician, Skip Garibaldi (yes, the guy who disproved Garrett Lisi’s TOE), who said this: “Find a drawing in which the jackpot is unusually large and the number of tickets is unusually low.”

Argh! It’s so simple! For example, the March 6, 2007, Mega Millions drawing reached a record $390 million; 212 million tickets were sold. Elaine and Barry Messner, of New Jersey, split the pot with truck driver Eddie Nabors, of Dalton, Georgia, who, when asked what he would do with the money famously said, “I’m going to fish.”

But it was a bad bet.

Despite the massive prize, the huge number of tickets sold meant that a dollar spent on this lottery returned only $0.74 (versus $0.95 for roulette). In fact, Mega Millions and Powerball have never once been a good bet: Extreme jackpots generate extreme ticket sales, increasing the chance of a split pot – the average return on a one-dollar Mega Millions ticket is only about $0.55.

“But state lotteries don’t get the same kind of press,” says Garibaldi. In rare cases, a state lottery jackpot will roll over a couple times without spiking ticket sales.

Here’s the formula for finding a good lottery bet: Look for an after-tax, cash value of the jackpot that exceeds 0.8 times the odds against you, and in which the number of tickets sold remains less than one-fifth this jackpot.

If you happen to be away from your spreadsheets, here’s how to approximate the formula: Look for a jackpot that’s rolled over at least five times and that remains below $40 million. It’s a good bet that it’s a good bet. And by a good bet, I mean a positive expected rate of return – over time, a dollar invested returns more than a dollar. To wit: a $1.00 ticket for the March 7, 2007, Lotto Texas drawing had an expected rate of return of $1.30. That’s a darn good bet.

Take a minute to scroll through online lottery listings till you find one that meets the criteria for a good bet. Okay, so you finally found one – what now?

Pick the most unpopular numbers, that’s what. By playing unpopular numbers you won’t win any more or less often, but you’ll less often split the pot with other winners.

Don’t pick the number one. It’s on about 15 percent of all tickets. Similarly, avoid lucky numbers 7, 13, 23, 32, 42, and 48. Better are 26, 34, 44, 45, and especially overlooked number 46. Avoid any recognizable pattern, but give slight preference to numbers at the edge of the ticket, which are underused. In mathematical terms, picking a unique ticket makes the jackpot look bigger and thus your lottery dollar look smarter.

If players in a 1995 UK National Lottery drawing had played unpopular numbers, they might’ve avoided splitting a £16 million pot 133 ways. That’s right – 133 people picked the numbers 7, 17, 23, 32, 38, 42, and 48, all straight down the ticket’s central column. Each got £120,000.

But despite your newfound ability to punk the lottery, the moral of Garibaldi’s surprisingly accessible paper on the subject is that while you can frequently make the lottery a good bet, it’s almost never a good investment. With money spread like a bell curve across different risk profiles according to the widely used portfolio theory, the extreme risk of a lottery means that in order for the left, risky edge of the bell curve to be a dollar tall, the total area of said curve has to be, like, $10m.

Play smart over enough drawings, and eventually you’ll win more than you spend. But unless you can buy all the tickets (another fun and somewhat involved statistical story!), you’re more likely to run out of money first.


Garth Sundem About the Author: Garth Sundem is a TED speaker, Wired GeekDad, Wipeout loser and author of books including Brain Trust: 93 Top Scientists Reveal Lab-Tested Secrets for Surfing, Dating, Dieting, Gambling, Growing Man-Eating Plants and More. He lives with his wife, two kids and two Labradors in Boulder, CO, where he just finished interviewing over 130 Nobel, MacArthur and National Medal of Science winners while sitting in the backyard garden shed. Follow on Twitter @garthsundem.

The views expressed are those of the author and are not necessarily those of Scientific American.

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  1. 1. nigel_turner 1:39 pm 03/9/2012

    Although it is theoretically possible to find a single lottery that has a prize that is large enough to have a positive expected return, the simple fact is that the larger the prize, the more players buy tickets nullifying any positive expected return (see Turner & Ferentzy, 2010, International Gambling Studies, 10(1), 19–30). The fact is that in reality the money paid back to the players for lottery tickets is typically less than 50% of the money taken in. Also there are enough players who believe in strategies such as betting on unpopular number so that they sink their life savings into it and end up losing. In economics this is called an efficient system — any “edge” possible once discovered will quickly spread and therefor be nullified. And gamblers spread beliefs about an edge because its much easier to make money telling people how to win, than it is to make money from playing the lottery. The fact is that the difference between the most popular and the least popular numbers is often too small to provide an edge to the player who chooses unpopular numbers. I published an article (Turner, Journal of Gambling Studies, 2010, 26, 421-439) that examined actually lottery draws that demonstrated that there is no ticket preference strategy that can provide a player with a long term positive expected return. The original article by Aaron Abrams and Skip Garibaldi is appropriately cautious about its conclusions and discourage lotteries as an investment, but it does encourage the belief that there are strategies that suggest when a lottery is a good bet (but bad investment). I note that neither this summary or the original article by Aaron Abrams and Skip Garibaldi mention that the people who are most apt to fall pray to such information about being able to beat lotteries by (e.g., by playing unpopular number for example) are problem gamblers. Papers such as this may unintentionally encouraged the delusions of people who already have a serious psychiatric problem, or non-problem players who may be at risk for a problem, to seek out a mythical strategy that is doomed to fail and may lead to a serious gamblng problem. The original article does draw a correct conclusion, that the best way to guarantee winning from a lottery is to own the lottery — and that is precisely why lotteries are used as a means of collecting “voluntary” tax from the poor (at math).
    Nigel Turner, Ph.D.
    Scientist, Centre for Addiction and Mental Health

    Link to this
  2. 2. sdoig 2:46 pm 03/14/2012

    What Dr. Turner said. But if you must play, let the lottery machine pick random numbers for you; little chance of a pattern there. Finally, re the “buy all the tickets” strategy, it can’t physically be done without tying up hundreds of lottery machines all week; you can’t walk into the lottery office with a big check and buy all combinations at once. But even if you could, there’s a huge danger. Let’s say the Powerball game goes to $250 million, and you could spend $146,107,962 to buy all combos. The balls drop, you have the winner — but so does someone else. You get $125 million and thereby have lost more than $21 million. (Actually, even worse is the fact that you only get about half the jackpot if you want it as a lump sum payout instead of a 20-year annuity.) Bottom line: The lottery is a cheap way to dream about becoming suddenly wealthy, but there’s no strategy that makes it a good bet, much less an investment.

    Steve Doig
    Cronkite School of Journalism
    Arizona State Univesity

    Link to this
  3. 3. Jape77 3:49 pm 03/29/2012

    My brother the statistician puts it much more succinctly: “Random numbers are like honey badger; they don’t give a shit.”

    Link to this
  4. 4. 2:20 am 10/12/2012


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