Lottery Math

written by bryan on March 29th, 2007 @ 06:23 PM

It’s been said that lotteries are “taxes on those who don’t know math”. If you buy a ticket for every draw, on average, each 2$ 6/49 ticket sold is worth 94 cents. The lottery commission gives out 47 cents on each dollar, so the math is pretty easy.

But the 6/49 has an interesting property: if the jackpot is not won, it rolls over to the next draw. So they pay out more than 47% of the take on draws where a really big jackpot is won. So this leads to the question: is it worth buying a ticket for these really big jackpots?

They tell us what the jackpot is, and they even tell us what the odds are, but they don’t calculate the expected returns for us. That’s because to calculate those numbers you need to know the number of tickets sold, and even the lottery commission doesn’t know those numbers until the draw date. However, we can guess based on previous draws. The only numbers I found were for the $54 million dollar draw (50 million tickets) and for the preceding 30 million dollar draw (27 million tickets). Source. If anybody has any other sources, let me know!

I punched the numbers for those two draws, and have published the results. For the 30 million dollar draw, the expected payout on a $2 ticket was $1.62. Not quite worth it yet, but a significant step up from the normal 94 cents. However, on the $54 million dollar draw, the payout per ticket was an astounding $2.94! The lottery commission actually lost money on that draw, but that was more than compensated for by the huge number of losers on the previous draws where the jackpot wasn’t paid out.

One interesting note is how the payouts change when a 30+ million dollar prize is unclaimed. ( Look at the bottom of this page ). I’m not sure why they do this, but I guess it’s to make it less obvious that the odds are usually in your favour on these draws. For instance, if they wouldn’t have done this on the $54 million game, it would have actually been a $109 million dollar jackpot. And since they only sold $100 million dollars worth of tickets for that draw, it would have been obvious from the press release that the odds were actually in the player’s favour, for that draw only. They want things to look good, but not too good; there are enough problems with gambling in Canada as it is. Overall the house is taking 53 cents on every dollar, and that hasn’t changed.

Is it a Good Idea to Join a Group?

Group lottery ticket buys are very common. Everybody in the office pool throws in $2, a group of tickets are bought, and you split the winnings if you win. This generally increases the value of the “dream”: it turns it into a shared dream, something to talk about. They can really be a lot of fun. Usually these pools buy one ticket per person, but that’s not really necessary, given the above principle of buying as few tickets as possible. 20 people could throw in a dime, and you could still talk over coffee about what you’d do with your share of the winnings. Your office pool dream cost less than the office pool coffee!

There’s another good reason to join a group. Most people who win give some of their winnings to friends and family. I’d do that, too. But if you’re going to share the winnings, shouldn’t you share the cost, too? Get your friends and family to join your buying group!

My Modest Proposal

Taking all these principles into account, I’m proposing a buying group. The group will buy a single ticket for every lottery where the odds are close to even. I’ll set that at $30 million and above. It’ll cost $2 to join the group, and the group will keep going until it’s out of money. $5 and $10 wins will be reinvested, everything else will be paid out on a pro-rata basis. I’ve already purchased a ticket to Saturday’s draw. If you want in on it, just e-mail me. If I know you, I’ll take your word that you’ll eventually send me the $2. Otherwise, you’ll join the group as soon as I get the $2.

The spreadsheet with my calculations is available on Google in a read-only format. If you want to double check my formulas or play with the numbers, you can download the spreadsheet exported to Excel format.

Good luck!


  • Wayne on 29 Mar 19:49

    Whether it is a rational bet or not is not the point:

    $2 buys you very long odds at a lot of money $2 buys you very long odds at 2 or 3 times a lot of money

    In other words, it’s stil $2 that buys you the hope of being rich.

  • Jason on 29 Mar 23:00

    Count me in. I suggest we use the numbers “1 2 3 4 5 6”.

    The other reason to join a lottery pool is to minimize future regret. Wouldn’t you feel like a sap if all your coworkers got rich and quit, and you hadn’t joined the pool.

    Maybe buying lottery tickets makes sense if the utility curve for money is s-shaped. Losing a dollar to the lottery now and then doesn’t matter much, but winning a million dollars would radically change your life, much more than a million times the trivial loss of a dollar does. But then eventually more money doesn’t matter much.

    Anyways, Wayne is probably right—the dream of being rich is worth more than $2 to many people.

  • Bryan on 30 Mar 08:04

    Actually, “1 2 3 4 5 6” is a poor set of numbers. They’re just as likely to be drawn as any other number, but we also want to minimize the number of people that we share the prize with, and I’m sure there are more people out there with the same idea as you had.

    Similarly, avoiding the numbers less <=31 and especially <=12 might be a good idea because of people’s prediliction to pick dates.

    For Saturday, I used Quick Pick.

  • Bryan on 01 Apr 06:31

    Not a single number matched for the March 31 draw. Nobody else won the jackpot, so we’ll try again on the 4th…

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