# Lottery - Probability

Looking over Canada's Western Lotto Max and Daily Grand lottery probabilities, a combinations calculator shows the actual chance of getting 7/7 with 50 numbers is 1 in 99,884,400. However, that's if you only had 1 selection per ticket. Since there are 3 selections per ticket, their website accurately shows the chance is 3 times better (99,884,400/3 = 33,294,800).

There are also 14 additional draws to the main draw where you must match 7/7 for another chance to win a jackpot. Their website shows that winning 1 of the 14 additional draws has an equal chance of 1 in 33,294,800 just as the main draw. Now, this is where my question comes in: aren't the odds of winning the jackpot of 7/7 actually better than 1 in 33,294,800 considering there is more than 1 jackpot of 7/7?

With 15 draws (1 main draw + 14 additional draws), aren't the odds of winning 7/7 for any 1 of the 15 draws actually 1 in 2,219,653 (33,294,800/15)? The odds should be 1 in 33,294,800 if there was only one main draw, but with 14 additional draws, it seems like the number is wrong.

Also, the Daily Grand lottery shows that winning 5/5 from a range of 49 numbers per one ticket has a chance of 1 in 2,224,698. However, the combinations calculator shows the real chance is 1 in 1,906,884. It seems that matching 5/5 has a higher probability than what is shown on the official lottery website.

While their website is correct that matching a specific 7/7 combination has a chance of 1 in 33,294,800 - my question is whether matching 7/7 for any 1 of the 15 main draws has a probability of 1 in 2,219,653 (33,294,800/15) versus 1 in 33,294,800? Since it doesn't matter which of the 15 specific combinations you match, any 1 of the 15 has a better chance than the website claims, am I right?

• The odds given in the Daily Grand are correct. Note that the 1 in 2,224,698 odds are for matching 5/5 and not matching the Grand Number. – Daniel Mathias May 28 at 22:21
• I know nothing about the Canadian lottery system but as you have described, if you want the chance to win any of the $15$ draws then yes the chance is $1/2219653$... (Well strictly speaking you need a tiny teeny adjustment just in case some of the $15$ results are identical but lets ignore that.) BTW how does a lottery with $15$ draws work? Do they split the pot into $15$ equal shares and assign one share to each draw? – antkam May 29 at 4:57
• I see, then if you don't care whether you win the \$2m or \$1m prize, as long as you win any of the $15$ draws, then yes the chance is $1/2219653$ to a very good approximation. – antkam May 30 at 2:18

I will use $${n\choose r}$$ to denote the number of ways to choose a set of $$r$$ numbers out of $$n$$ possibilities.
Let $$A=$$ Prob of matching $$5/5$$ and the GN $$= {1 \over {49 \choose 5}} \times {1 \over 7} = {1 \over 1906884} \times {1 \over 7} = {1 \over 13348188}$$
Let $$B=$$ Prob of matching $$5/5$$ (don't care whether you match GN or not) $$= {1 \over {49 \choose 5}} = {1 \over 1906884}$$
Prob of matching $$5/5$$ and not matching the GN $$= B - A = {1 \over 1906884} \times {6 \over 7} = {6 \over 13348188} = {1 \over 2224698}$$ as the website claims.