Thursday, December 20, 2012

Lotteries over the Long Run

People who play the lottery generally know that their odds of winning are very low. However, some ticket buyers I’ve spoken to believe (or hope!) that they’re bound to win if they keep playing long enough. I decided to do some simulations of the Lotto Max lottery to examine this belief.

I ran a million simulations of playing 2 tickets per week for 25 years. At $5 per ticket, that’s a total cost of $13,000 for each of a million lottery players. I included all the gory details about how the prize pools are determined, winning a free ticket when matching 3 numbers, and everything else.

A simplifying assumption I made was to treat all chosen number combinations as random instead of having some of them chosen by people. I also had to make assumptions about ticket sales: I chose sales of $25 million per draw when the previous jackpot was won and sales of $15 million more than the previous jackpot when it wasn’t won. This crudely models the hysteria that comes with big jackpots.

The results were dismal. The median total winnings were $1840. This means that for the $13,000 invested, half of the million lottery players had total winnings of less than $1840 and half more.

How many of the lottery players at least made back their $13,000? Only one player out of 529. So 528 out of 529 lottery players would lose money over the 25 years.

How many lottery players hit it big with enough money to retire permanently ($5 million or more)? Only one player out of 10,000. In a town of 10,000 lottery players buying tickets for 25 years, 9999 of them would never realize their lottery-based retirement dreams.

How long would you have to keep playing 2 tickets per week before the odds of winning the big jackpot reach 50/50? The answer is about 1900 centuries!

I don’t expect to make a dent in lottery sales with rational arguments, but if even one person stops buying lottery tickets after reading this, I’d be happy.


  1. Hey Michael,

    Out of curiosity, how do you perform this kind of simulation?
    I think I remember you have a software background: do you program it? - or is there a tool?

  2. @Daryn: If a calculation is simple enough I use a spreadsheet, but in this case I programmed it in C++.

  3. C++, heh, you young lads with your new fangled programming languages, if you can't write it in Assembler or Forth it's not worth running!

    So what you are saying is Lottery tickets still are better than Nortel stocks were?

  4. @Big Cajun Man: If you kept the dividends over the years, Nortel stock may have been better than lotteries, but not by much.

  5. Michael,
    thank you for bringing up the probability numbers.
    My 2c:
    - low probability is not all: last spring while walking near Trafalgar Square in London, England I accidentally met the only couple I knew in London; probability of this is similar to winning a jackpot. Did not make me an avid lottery player.
    - I disagree with your call to stop buying tickets - it's a kind of tax on absence of intelligence and vent for a small time gambling craving. Collected funds are used for public good :)

  6. @AnatoliN: Presumably you saw many people in London as you moved around that day. Let's say that you saw 1000 people well enough that you would have recognized someone you knew. The population of London is less than 10 million. That puts the odds of your event at about 1 in 10,000. The odds of a particular ticket winning the Lotto Max jackpot is about 2600 times smaller than this.

    On the "stupid tax", I agree that this is what it is. But, I like people and would prefer to see them not waste their money. From a purely selfish point of view, much of my tax money goes to supporting poor people, who happen to be the ones who buy most of the lottery tickets. The net benefit of lottery tickets is negative.

    The best argument in favour of lottery tickets is that some people are going to gamble anyway and we might as well capture the profits for our government instead of foreign governments or organized crime. However, if this were the only reason lotteries exist, then there would be little reason to advertise them. We would spread the word just enough to make sure that gamblers who need an outlet know about the tickets, but we would not run massive slick advertising campaigns designed to ensnare as many people as possible.

  7. First let me say I cannot remember the last time I bought a latter ticket. But the "skewness" argument makes a lot of sense. You can argue that spending $2 a week on a lottery ticket is a trivial cost that will have zero bearing on your standard of living. Indeed, it might even give you a wee bit of enjoyment. If for that tiny cost you get even a remote chance of winning a fortune, could that not be considered a rational purchase?

    Of course, this does not apply to the guy I saw at Shoppers Drug Mart yesterday who was buying multiple lottery tickets with his Visa.

  8. @Canadian Couch Potato: It's true that some people buy lottery tickets just for a little fun with a trivial amount of money. But this seems to be a very small fraction of purchases. In a way this remonds me of the poker boom. Everyone seemed to be interested in playing. If you asked them whether they were driven by the prospect of making money or just the fun of playing, most would say either the fun of playing or both. However, most novice players figure out quickly that they can't win and they quit. Only a small minority of people stick to playing poker because they enjoy playing.

  9. @Anonymous: The following anonymous comment seemed to get lost:

    "Why don't you do a study on how many people starting with $5 a week invest their way to the equivalent of a lottery win or even a comfortable retirement, prefereably an early retirement.
    Even take away the $5 a week limitation."

    I used $10 per week in my post. With a 3% real return, the $10 per week (growing with inflation) would become about $19,000 (plus inflation) after 25 years. I'd rather have this money than the median lottery return of $1840.

  10. Making regular small bets for the possibility of an infrequent but large gain sounds an awful lot like what Talent does/did with financial markets. However, financial markets have tanked on average every 7-8 years so the chances of eventually profiting are much better than with lottery tickets. Then again, there is the interesting (but dated) story of the guy who cracked the code for scratch and win OLG.

  11. @Dan: I haven't read Taleb's latest book, but I intend to. Just about anything has to give better odds of profiting than lottery tickets. I remember that Wired article -- it's definitely worth a read.

  12. Sometimes I think those people who buy a lottery ticket on a regular basis should go take some poker lessons first before they go buy another one. Maybe that way they can weigh their odds a bit better.