Tuesday, March 24, 2009

Winning the Lottery

We’ve all heard that lotteries are a sucker’s game, and this is mostly true. Taking Canada’s Lotto 6/49 as an example, only 47% of the money spent on tickets gets repaid in prizes. So, players pay $2 for a ticket worth only 94 cents. However, there is a way to stack the odds in your favour if you have the right information.

There are any number of books and web sites touting systems for picking lottery numbers based on which numbers have been drawn more or less frequently in past draws. None of these work. My system of playing the 6/49 actually has a long-term expectation of profit.

To understand this system, we need to start with a little information about how the 6/49 works. Each player chooses any 6 of the numbers from 1 to 49 and pays $2 for this combination. There are just under 14 million combinations available. The draw consists of 6 numbers plus a bonus seventh number. The player gets prizes if his combination matches 3 or more of the drawn numbers. The more matches, the bigger the prize. 

But what happens if two players want to buy the same combination? The answer is that they can both have the same combination of numbers on their tickets. When you buy your ticket you aren’t buying exclusive rights to your combination. In fact, when jackpots get large, Canadians often buy more than 14 million tickets, which means that there have to be ticket collisions because there aren’t enough combinations to go around. 

So, if you’re lucky enough to have your exact combination drawn for the top prize, you may have to share the prize with other people who bought the same combination. A similar method is used for the other large prizes for matching 5 numbers plus the bonus number, matching 5 numbers without the bonus, and matching 4 numbers. A fixed amount of money is allocated to each of these pools and all the winning tickets share the relevant prize pool equally. 

This brings us to the main idea of choosing a combination that no other player has chosen. This won’t increase your odds of winning a prize, but it will increase the amount of money you win if your numbers do come up. In the case of the top prize, you would get the whole thing instead of sharing it. For the other large prizes, the number of winners sharing the prize pools would be slightly smaller. 

How much this helps depends on the size of the prize pools and the number of tickets bought. Generally, the bigger the prize pool, the more tickets get sold, and the bigger the advantage in buying combinations that nobody else has. 

According to the Wikipedia Lotto 6/49 page, the biggest jackpot was $52,294,712 on October 26, 2005. An estimated $99,400,000 worth of tickets were sold. From the Lotto 6/49 prize rules, this is enough information to calculate all the prize pools. 

For the average player, even if all the prizes for this draw were paid out, only about 76% of the ticket sales would be returned in prizes. So, a $2 ticket was worth at most $1.52. But what about unique tickets whose combination wasn’t bought by any other player? 

I decided to run some simulations. For this draw, about 49,700,000 tickets were sold. Because there are only 13,983,816 combinations, you might think that every combination would be purchased by at least one player, but this is exceedingly unlikely. In fact, if the ticket combinations are chosen randomly, the number of combinations not purchased is expected to be very close to 400,000. 

What if some lottery insider, Ian, could get this list of combinations not purchased, and buy all of them just as sales close before the draw? This is the scenario that I simulated. I assigned 49,700,000 tickets randomly, and assumed that Ian purchased any unused combinations. The simulation determined what Ian’s payback was for each possible set of numbers drawn. This gives us a distribution of possible results for Ian. 

It turns out that Ian purchased 400,154 tickets for $800,308, and his average total prize count is $1,820,000! Ian’s payoff is highly variable, but repeating this strategy reduces the risk. After 50 lotteries, Ian’s probability of at least doubling his money is about 60%. 

Of course, because Ian is getting more than his fair share of the winnings, everyone else is getting less. So, unless you’re an insider like Ian who can improve his odds, your best bet is to avoid lotteries. 

I’d like to think that lottery officials have measures in place to prevent this type of abuse by insiders, but given how long it took them to find and acknowledge the problem of retailers cheating the system, I’m not optimistic. It wouldn’t be hard to detect lottery abuse of this type if you think to look for it.


  1. Interesting post Michael. I do hope OLG bans insiders from buying tickets. Even if the win is legitimate, the optics just doesn't look good. That is one reason why most giveaways specifically bar insiders.

  2. CC: I hope lottery officials ban insiders as well. But they should also check for unusual ticket buying of combinations not already purchased as a check of whether the ban is effective.

  3. A Polish guy tried to game the Irish lottery in the 90s (it was 6/36 back then so all combinations could be purchased for $980,000 odd). The lotto tried to push back by making up rules about how many tickets you could buy in one location and so forth but he still made a 300,000 profit.

    I read (in a book about chinese food!) that the most popular numbers by far are 1,2,3,4,5,6 and 5,10,15... The lotteries in the US have special insurance against those numbers coming up because they'd be hit like you wouldn't believe with winners. As happened one weekend when multiple people won using six particular numbers printed on fortune cookies.

  4. Guiness416: If all the Polish guy did was buy all the possible combinations, that sounds like a good plan rather than gaming the system. He deserves his profit if he can legally exploit a government that doesn't know how to design a lottery. The 6/49 wouldn't have any problems if popular numbers get drawn because they would just divide the prize pool into smaller shares. So, people who choose popular numbers get smaller prizes if they win.

  5. Before I read Guiness416's comment, I thought that choosing strange combinations of numbers would be a good opportunity to choose unique lottery tickets, say 1-2-3-4-5-6, or 34-35-36-37-38-39.

    However, since as he said, 1 through 6 is a popular choice, that sequence wouldn't work.

    I intuitively know that sequential numbers aren't likely to win, but for some reason, it seems like 9-14-26-33-39-46 is more likely. I guess it's because it looks more random, but each set of six numbers is as likely to win, sequential or not. If everyone used quick picks, it would be impossible to try to pick rare sequences by reasoning alone.

    On a tangent, random.org generates what the site claims are true random numbers based on atmospheric noise. The site has some random number generators for lotteries of the world. You can also use it to simulate electronic coin flips, or shuffle electronic playing cards.

  6. Gene: It's true that every combination is equally likely even if it doesn't seem that way. I wonder if the quick picks are designed to be uniformly random or if they avoid combinations that don't "look" random.

  7. That sounds like a good strategy - of course you need the info, the $400k and the means to buy all those tickets at the last minute....

  8. I wonder how many tickets are picked by a person compared to the quick picks. Since people don't pick numbers randomly an insider who had access to the statistical correlation of customer picks from past drawings should be able to select less common number combinations giving themselves some advantage over the average player's odds.

  9. Jordan: Using statistics on player tendencies would make it possible to get better odds of winning than the average player, but I doubt that this would be enough to develop a profitable strategy. The long-run average return on investment for the Lotto 6/49 is a 53% loss. Anything short of certainty about which combinations have been purchased makes it hard to overcome such terrible odds.

  10. The interesting thing is that the lottery will provide you with a list of all the winning numbers for 6/49 and if I remember correctly 6 sequential numbers has never won before. Still does not change the odds though.

  11. Michael C: There are only 44 ways that 6 consecutive numbers could come up. The odds of this happening are 1 in 317,814. So, it's not surprising that it hasn't happened yet.

  12. To guinness416

    How I can crank the same on Australian Powerball results
    Please email me if you can do it. Thanks!