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.