# Probability of Winning Lotto Max (Canadian Lottery)

TR Smith is a product designer and former teacher who uses math in her work every day.

In Canada, Lotto Max is a uniquely structured lottery with a prize cap of 50 million dollars. If the jackpot exceeds 50 million, the excess amount is awarded in parcels of one million dollars during the "Maxmillions" draws. For example, if the Lotto Max jackpot reached 56 million dollars, there would be one main drawing for the 50 million dollar prize, and then six Maxmillions drawings worth one million apiece.

During each drawing, officials randomly select seven distinct numbers from 1 to 49, and then another bonus number. You do not select the bonus number on your ticket, you only select the seven main numbers.

A Lotto Max ticket costs \$5 but gives you to three different seven-number selections. One of the number sets you are allowed to choose yourself, while the other two sets are randomly generated by computer. You can think of the actual cost of a single selection as \$5 ÷ 3 = \$1.67, which is less than the \$3 cost of a single-play Lotto 6/49 ticket. The price is the same in every province and can be verified on each province's lottery website. This aspect of Lotto Max is confusing for some players who think you only get a single play for \$5; it is all too easy to forget about the other two computer-generated plays.

The main prize is awarded to the ticket holder(s) who matches all seven main numbers. Lower-tier prizes are awarded for matching 6/7 plus the bonus, 6/7 but not the bonus, and other partial matches. There is no special prize for getting 7/7 plus the bonus since you do not select any bonus numbers on your ticket. In order to win a Maxmillions runoff drawing, you must match 7/7 on the additional drawings. There are no lesser prizes awarded for partial matches on a Maxmillions runoff drawing.

## Odds of Winning Lotto Max

The odds of winning the main jackpot depends on the total number of possible lottery ticket combinations. Since Lotto Max is structured as a 7 out of 49 game, the number of different combos is

[49*48*47*46*45*44*43]/[7*6*5*4*3*2*1]

= 432938943360/5040

= 85,900,584

This is about 6 times the number of possible tickets in Lotto 6/49. The odds of a single selection winning the Lotto Max jackpot are 85900584 to 1, however, since each ticket gives you three picks, the true odds are 85900584 to 3, or equivalently 28633528 to 1.

In probability terms, the likelihood of having a 7/7 match on your Lotto Max ticket is 0.000000034924. As a percent, this is 0.0000034924%, about half as likely as winning Lotto 6/49. This probably explains why in Canada, Lotto 6/49 is still the more popular of the two lotteries.

## Manual Selection or Quick Picks?

When you buy a Lotto Max ticket, you can either let the computer choose all three seven-number selections for you at random, or you can choose one of the selections yourself and let the machine pick the other two. You do not have the option to choose the numbers for all three selections yourself.

## Lesser Lotto Max Prizes and Their Odds

The jackpot is 87% of the prize fund. The rest of the prize fund is awarded to ticket holders whose numbers partially match the winning numbers. Here is a table with the odds of winning lesser prizes in Lotto Max drawing.

Partial Match
Likelihood
What You Win
6/7 + bonus
1 / 4090506.5
your share of 4% of the prize fund
6/7
1 / 99768.7
your share of 4% of the prize fund
5/7
1 / 1583.96
your share of 5% of the prize fund
4/7
1 / 71.6
\$20
3/7 + bonus
1 / 77.08
\$20
3/7
1 / 8.42
free ticket

In Canada, lottery prizes are tax free and can be received in installments or as a lump sum. The odds of winning a Maxmillions runoff drawing are the same as the odds of making a 7/7 match on a 3-play ticket. The more Maxmillions drawings there are, the better your chances of winning. If there are 10 runoff drawings, your chances are 10/28633528. But if there happen to be 47 runoff drawings, your chances increase to 47/28633528.

## More in This Lottery Series

How to calculate your odds of winning a prize in other lotteries.

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• jordan 3 years ago

does having more maxmillions increase your odds? A friend told me because this week has 50 maxmillions that your odds of winning go from 1:28.6 million to 1: 572 670. I don't understand how having more draws would increase your odds? Could you provide any clarity?

• Author

TR Smith 3 years ago from Eastern Europe

Thanks for the great question Jordan. Having more Max Millions draws for the excess money in pool doesn't increase your chances of winning the main prize (whose value is capped, thus creating the excess), but it does increase your overall chances of winning any prize.

Say you bought a ticket and your numbers didn't match the main prize drawing, but there was enough excess for one or more Max Millions drawings. For each of these drawings they select an entirely new set of numbers at random, so you still have a chance to win some of the excess, which is parceled out by the million. The more of these drawings there are, the more chances you have to win one of them.

• jordan 3 years ago

Thanks for the reply. This is where I get confused. So when I buy my 5 dollar ticket my odds of winning the jackpot is 1:28.6 million. If they are doing separate draws for all of the maxmillions draws how do your odds become better? I could see your odds increasing a bit because of more maxmillions but I cannot see the odds being 50 times better. Is it a lower ratio or do you just divide the 28.6 million by the amount of maxmillions?

• Author

TR Smith 3 years ago from Eastern Europe

Your chances of winning any particular drawing are still abysmal, but if you get N tries, your chances are N times better. Say I pick a number from 1 to 10 and you have to guess it. If I give you one guess, your chances are 1/10 = 0.1. But if I give you two guesses your chances are 2/10 = 0.2. (If I gave you 10 guesses, your chances of getting it would be 10/10 = 100%.)

The probability of matching 7/7 with a three-selection ticket is so minuscule to begin with that multiplying by N doesn't do all that much to improve your odds. But it is some improvement.

One other thing to consider is that when ticket sales reach the volume necessary to create a lot of runoff maxmillions drawings, it means the chances that some people bought the same numbers also increases. And according to the lottery website, if more than one person holds a matching ticket for a max millions drawing, they have to share that \$1,000,000.

• smarter then the avg bear 2 years ago

when the jackpot surpases the population of canada, why don't they just give everyone one million dolllars?

• Author

TR Smith 2 years ago from Eastern Europe

Well, the jackpot has never surpassed 35,160,000 times \$1,000,000 = \$35,160,000,000,000, but I suppose when the jackpot goes over \$35,160,000 they could give everyone a loonie (Canadian \$1 coin).

• Captain 3 months ago

I understand the fractional chance of winning a lottery as the product of the multiplication of the chances of picking individual numbers, less the one that has been picked, if any. So 48 * 47 * 46 *... . Why does this product then have to be divided by the product of (7 *6 *5 * ...)?

• jak 3 months ago

@captain -- if you work it out like 49*48*47*46*45*44*43 then you are counting a combination like (1, 3, 14, 17, 23, 26, 33) as something distinct from (33, 26, 14, 1, 3, 17, 23) because your way of counting it counts not just all the different ways of picking 7 numbers out of 49, but also all the ways of ordering those number. your method is counting 49 choices for the first number, 48 choices for the second number, 47 choices for the third number and so on.

but when you buy a ticket here in canada it don't matter what order the numbers are in. whatever numbers you pick they automatically put in numerical order on your receipt. there are 7*6*5*4*3*2*1 ways to order 7 different number, so you divide 49*48*47*46*45*44*43 by 7*6*5*4*3*2*1 to get the actual number.

maybe the lottery were you live is done differently and you have to get the numbers right AND get them in the right order.

• Molière 3 months ago

I'm not saying this explanation is wrong, I just don't understand why they give you 3 selections when you buy a ticket and make it so expensive. I would rather pay \$1.66 to play 1 set of numbers than be forced to pay \$5 to play 3 sets. If I'm understanding it correctly, giving you 3 selections effectively increases your odds of winning by a factor of 3. Well, why don't they make Lotto Max in such a way that with a single combination you get the same odds? I'm not so good with this kind of math, but would something like picking 7 out of 40 work out to be the same? That's why I never play this lottery, it seems like a scam to me.

• Author

TR Smith 3 months ago from Eastern Europe

Hi Captain, thanks for the question. Jak's explanation of why you need to divide by 5040 is correct. Thank you Jak.

• Author

TR Smith 3 months ago from Eastern Europe

Hi Moliere, thanks for your comments. I do not know why they set up Lotto Max to sell you 3 tickets for \$5, but I suppose they did some market research and determined they would get more sales with that strategy than with selling single tickets.

If there was a pick 7 out of 42 lottery game, the probability of winning with a single ticket would be 1/26978328, which is close to the probability of winning Lotto Max 1/28633528.

• Aldz 2 months ago

Hi Moliere,

The reason why they force you to by 3, is to prevent the lottery from being stolen by someone who buys every single combination. Remember, even when you personally select one set of numbers, the second and third sets are randomly generated. This makes it harder for you to purchase every single combination of numbers, as you are essentially forced to buy 3x the number of tickets. In that way, it is never going to be worth it to buy every single ticket combination, as it will always exceed the value of the main prize plus the value of all the maxmillion prizes.