Odds of Winning a Canadian Lottery
What are your chances of holding a winning ticket?

The Lotto 649 is a Canadian lottery where six different numbers are randomly drawn from a set of balls with numbers 1 through 49. What are your chances of holding a winning ticket?

The number of different ticket combinations (where you can’t use the same number twice)
can be determined by using:
49C6 = 13,983,816

So you have one chance in 13,983,816 of holding the winning ticket. The probability is 0.0000000715.
The more tickets you buy, the greater your change of winning... Suppose you buy 50 tickets (which will cost you \$50 Canadian). Your chances of winning go up - they’re now a whopping 50 out of 13,983,816 - for a probability of 0.000003575 (but not much of an improvement).

What about the probability of winning anything?

1. The probability of matching NONE of the six numbers is:
43Cx 6C0  over  49C6
where  43C6 is the number of ways of choosing from numbers not on your ticket;
6C0 is the number of ways of choosing the six losers on your ticket;
49C6 is the number of ways to choose 6 balls

which works out to be 0.436, or 43.6%... which means that about half the time, NONE of your numbers will match!

2. Similarly, the probability of matching ONE of the six numbers is:
43Cx 6C1  over 49C6
which works out to be 0.413, or 41.3%... the probability of matching ONE number!

3. The probability of matching TWO numbers is:
43Cx 6C2  over 49C6
which works out to be 0.132, or 13.2%... the probability of matching TWO numbers!

None of the above 3 results pays anything at all (i.e. matching none, 1 or 2).
If you add the 3 probabilities, you get 98.1%
Which means that 98% of the time, you won’t win anything!

The first winning ticket is the one where you match THREE numbers. The probability of this occurring (using the method above once again) turns out to be 0.018 or 1.8%... or about 2 out of 100... or about 1 out of 50. This means that you must spend about \$50 (on average) to hope to win \$10.

Getting the picture yet? Your chances of winning the big prize are next to nothing, and the chances of winning even \$10 are only one in fifty... Yet people win all the time because millions of tickets are sold every week - and somebody has to win the prizes. It’s just not very likely that it will be any one person in particular - and that includes you!

Words of encouragement:
- If you play Lotto 649 twice a week, every week, for the next thousand years, your chances of winning the big jackpot are still only about one percent!
- Cheer up - the odds of winning the Lotto Super 7 (7 numbers, 1 through 47) are 62,891,499 to one!

Winning and losing at the lottery - gambling odds stacked against you