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No Matter How Huge, Mega Millions Jackpot Will Always Be a Bad Bet

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

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lottery ticket for mega millionsYesterday my father-in-law asked me to buy him $100 in lottery tickets. He is ordinarily the kind of guy who would cite the quip “the lottery is a tax on people who can’t do math,” but these are not ordinary times. On Friday night the Mega Millions multi-state lottery will offer a $500 million jackpot, give or take, by far the highest jackpot ever offered in the history of the known universe. The prize is so high it exceeds the number of possible number combinations on a ticket, which is about 176 million. (In other words, the chance that any particular ticket is a winner is about 176 million to one.) The math seems to imply that a $1 ticket has an expected value of $500 million divided by 176 million, or nearly $3. Yet a closer look at the math reveals that the Mega Millions jackpot is a bad bet no matter how large the prize.

The reason? For starters, even if you hit the jackpot, you may have to share it if other ticket holders played the same numbers that you did. Indeed, this is the fate that usually befalls winners of big Mega Millions jackpots. Each of the three previous three highest Mega Millions drawings were won by two different ticket holders who had to split the jackpot; the fourth-highest drawing was split amongst four winning tickets.

Certainly, the threat of having to split is there, but does that really make it a bad bet—especially when the jackpot is so very high? According to the mathematicians, yes. As the number of tickets sold goes up, the chance that more than one person will share in the jackpot does as well, according to a well-known mathematical function called a binomial distribution. When Emory University mathematicians Skip Garibaldi and Aaron Abrams worked through the equations, they found that lotteries are generally a terrible bet—Mega Millions and Powerball particularly so. (I encourage you to take a look at their paper “Finding good bets in the lottery, and why you shouldn’t take them,” which was published in the American Mathematical Monthly in 2010.)

Even in the case of the current drawing, which offers a jackpot so large that Garibaldi and Abrams show how it should only occur on average every 22 years, the number of tickets that go out is correspondingly large. “I ran the numbers last night,” Garibaldi told me over the phone. “You can tell by the amount they estimate the jackpot to be what they estimate the ticket sales to be.” Based on the current jackpot, an estimated 380 million tickets have been sold this week. The estimated return on an investment of this week’s Mega Millions drawing? Negative 19 percent, per his calculations.

Still, even if that number was positive (and it was, once, in a Texas lottery), buying lottery tickets will never be a sound investment strategy. Using modern portfolio theory, the authors show that the chances of winning the lottery are so incredibly low that the risk precludes any wise investment, no matter what the expected rate of return.

That said, Garibaldi bought five tickets. “I know it’s throwing money away,” he admitted, but it’s fun to be in on the action. That’s why I don’t feel bad about buying 100 tickets for my father-in-law, either. Besides, he promised to split the jackpot with the rest of the family. With zero out-of-pocket expenses, my rate of return is infinite.

Image by vvvracer on Flickr

About the Author: Michael Moyer is the editor in charge of space and physics coverage at Scientific American. Follow on Twitter @mmoyr.

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

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  1. 1. Tucker M 10:45 am 03/30/2012

    There’s another way to look at the math (this is a bit tongue-in-cheek, but stay with me). For most people, the price of buying one ticket is effectively zero, in that saving that dollar instead of spending it isn’t going to change your life one whit. On that theory, it is irrational NOT to play the lottery; you can essentially get a (vanishingly small) chance at vast riches for free, so why not take it?
    Of course, your chances of winning are effectively zero, too. That’s why I never spend more than $1; doubling a zero chance still nets you a zero chance. So I say: while you gotta be in it to win it, don’t spend more than a dollar. That’s just irrational. :-)

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  2. 2. @michaelmccutch 11:24 am 03/30/2012

    Additionally, research shows that when we buy a lottery ticket and dream about what we’d do with the winnings the same chemicals are released in the brain – albeit a much smaller amount – as if we were to actually win.

    So either way, with $1, we’re buying a nice little mental effect regardless of the outcome.

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  3. 3. bigbopper 11:37 am 03/30/2012


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  4. 4. andries1+ 2:23 pm 03/30/2012

    I play for the following reasons: you can’t win if you don’t buy a ticket, my ticket’s chance to win was exactly the same as the chance was for the one that did win and (most importantly) if my usual numbers won and I did NOT buy a ticket, it will be a bit of a disappointment…

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  5. 5. scottlmoore111 4:08 pm 03/30/2012

    Regardless of the odds someone WILL win and I don’t care squat about a $1.

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  6. 6. captain_walker 4:23 pm 03/30/2012

    Those who take a chance may have, say, 1/200,000,000 of a chance of winning.

    Those who do not take a single chance have absolutely zero chance of winning – which is much smaller than 1/200,000,000.

    Therefore those who play have 1/200 million divided by zero, greater chance of winning – isn’t that correct? What’s that number called i.e. the result of dividing a very small number by zero? I think it might start with an ‘I’. :)

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  7. 7. GG 4:29 pm 03/30/2012

    This is when some math knowledge will turn you into a dumbo. Here are some pointers:
    1) Someone WILL WIN, sooner or later. If you don’t play, you don’t count at all.
    2) Your financial loss from playing is practically zero. You can just skip your morning coffee, to make up for that money.

    If I win, I will commission a book on countering all the pseudo-scientific fallacies being spread and published by the science establishment (you thought only religious people can be delusional? think again)

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  8. 8. Geopelia 5:40 pm 03/30/2012

    Here in New Zealand, we can buy Bonus Bonds. Instead of paying interest, they give the chance of prizes.
    The prizes are much smaller than Lotto, but in the end we get all our money back.

    Lotto tickets have become too expensive now for a regular weekly purchase, but there are still queues for the big jackpot tickets.

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  9. 9. woodswoman1 6:37 pm 03/30/2012

    I believe that whenever a jackpot exceeds 100 mil, the remainder should be split into 1 million dollar prizes to be drawn along with the 100 mil draw. Just think, 200 or 300 more people could be winners of 1 mil dollars. What fun that would be and I believe it would give the economy a boost in a small way. More people would buy tickets if they knew they had a better chance to win just 1 million.

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  10. 10. Durazac 7:30 pm 03/30/2012

    I never buy em’, but I certainly like a lot of excitement for a little bit of pocket money. In the words of this workin’ man, it seems like some real cheap brain candy.

    c’mon – over a half a billion smackers!

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  11. 11. ridelo 4:51 am 03/31/2012

    If you’ve the ‘luck’ to win that amount of money and if you’re not accustomed to handle such sums, chances are that you’ll never sleep quietly any more with so many sharks around you.
    No offence to real sharks!

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  12. 12. jtdwyer 7:15 pm 03/31/2012

    GG – Well put! “Mega Millions jackpot will always be a bad bet” is not always true.

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  13. 13. Johnay 12:29 pm 04/1/2012

    I see comparisons between the odds of winning per ticket (very small but non-zero) vs the odds if you buy none (zero), but are your odds actually zero if you buy none? Given how past winners have divided their winnings among relatives and such who did not buy their ticket with them, what are the average odds of someone who did not buy a ticket getting a payout of some sort? I imagine it would increase with the overall number of people playing as a percentage of the population. Also that would apply to everyone, players included, so I guess you have to think of buying a ticket as increasing already non-zero odds. But of course all that is based on having no knowledge of whether generous friends or relatives are playing or how many tickets they have bought. If you know those things you can calculate your odds more precisely.

    As for value for your money, don’t forget that what’s collected and not paid out in prizes (& lottery-running overhead) is put toward funding various public services, so on the balance you’re only losing the overhead which you can view as a very small entertainment fee.

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  14. 14. Jehovah Akbar 5:34 pm 04/4/2012

    Virtually all of us share the same feeling when we buy a long odds ticket. A genuine overly optimistic feeling I am going to win.

    I gave up thinking about why people think they will win when I came up with an answer that fits. I write this now in part to see if someone has a better psychological theory.

    Have you ever calculated the odds of your existence? To exemplify, going back through just 1000 generations of your paternal lineage, if any of the men had delayed or hastened ejaculation by a second or two you would not have been born.

    I have not calculated the probability of your birth but it must be somewhere in the range of the probability of this universe or any universe capable of evolving sentient beings to talk: one in 10 to the 243rd power (Rees Just Six Numbers).

    Odds of 176 million to 1 feel quite achievable considering each of us have won a birth lottery that make those odds look like a ‘sure winner’.

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  15. 15. aksal 11:54 am 08/1/2012

    I use smart selection from I think your chances get better if you select 20 – 25 numbers and get your lucky combination using those numbers only. If your selection is right your chances are much higher.

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