Can someone explain why the probability of matching 2 numbers is stated to be equal to 1:210?

There are 42840 different combinations of three numbers between 1-36 (36*35*34).

Now in order to match 2, I have 3 ways of choosing 2 out of the 3 actually picked numbers and 33 ways for the third number that is not part of the ones that get picked. And there are 6 permutations for each of these 3*33 triplets that have 2 right and 1 wrong.

So I end up having 6*3*33=594 ways of matching two (33 of which actually have the first two numbers right in the order they are drawn). There are only 6 more triplets that have all three numbers right.

And if I divide 594 (or 561 or 600) by 42840 it yields a likelihood that's different from 1:210. The other probabilities on the raffle page make sense but this one doesn't (for me at least).

So my question here is: Where's the flaw in my math and how's this probability computed? I assume they use (3*2*34)/(36*35*34)=1/210, but aren't the more than 3*2*34 = 204 variants that match two?

Thanks!

That's the correct math. There's only 204 variants that match 2 on any given 3-number combination. Theres 3 numbers I can pick from, then 2, then any of the 34 left.

/e Also, keep in mind that there is the "match 2 ordered" which only applies to the first 2 numbers, ordered, so we kinda have to take this in consideration when saying 1/210 chance of that specific win, if you add match 2 ordered + match 2 you have more than 204 possible hits.

This post was edited by Cersei on Sep 20 2022 02:00pm

it's a hypergeometric distribution so

Let's presume 1 2 3 get drawn (in this order).

I'll buy one ticket and want to get two right and one wrong. I'll win if I fall into any of the following cases:

I: 1 and 2 and any of the 33 wrong numbers. So 33 triples. This times 6 for permutations yields 33*6=198.

II: 2 and 3 and any of the 33 wrong numbers. So 33 triples. This times 6 for permutations yields 33*6=198.

III: 1 and 3 and any of the 33 wrong numbers. So 33 triples. This times 6 for permutations yields 33*6=198.

And this adds up to 594. Again there are 33 among those that have the first two correct and ordered. So there's 561 triplets to win 2 matched, unordered.

EDIT:




The left side is correct, but the result on the right side is wrong. I think it should be 33/2380, which is the same value I got.



This post was edited by TheOnlyDenny on Sep 20 2022 02:22pm

yup you right, confused my nPr button with nCr on my calculator
so they should probably change it in their raffle description

Are you guys accounting for the removal of the first number matched from the draw pool?

They also adjust the payout to the likelihood of winning (2, 3, ordered, unordered). So technically the amount won would have to change also (decrease in this case).

Here's an example why I say it is more than 204:

Imagine it is 1-2-3 that gets picked. How many variants are there that match two?

Case 1: We get 1 and 2 right and one wrong, which could be

1-2-4, 1-2-5, 1-2-6, 1-2-7, 1-2-8, 1-2-9, 1-2-10, 1-2-11, 1-2-12, 1-2-13, 1-2-13, 1-2-14, 1-2-15, 1-2-16, 1-2-17, 1-2-18, 1-2-19, 1-2-20, 1-2-21, 1-2-22, 1-2-23, 1-2-24, 1-2-25, 1-2-26, 1-2-27, 1-2-28, 1-2-29, 1-2-30, 1-2-31, 1-2-32, 1-2-33, 1-2-34, 1-2-35, 1-2-36.

That's 33, but for each these we can permute them in 6 different ways. For example: 1-2-4, 1-4-2, 2-1-4, 2-4-1, 4-1-2, 4-2-1. So there's 33*6=198 variants in this case (33 of which have the feature that the first two numbers are in correct order, 165 have two correct but in wrong order).

None of these contain the number 3.

Case 2: We get 1 and 3 right and one wrong, which could be

1-3-4, 1-3-5, ......, 1-3-36.

Adding permutations we, once again, end up with 33*6=198 variants. This time, since none of these contain the number 2, it is always unorderd.

Case 3: We get 2 and 3 right and one wrong, which could be

2-3-4, 2-3-5, ..., 2-3-36.

Plus permutations. As in case 2: 198 variants. None of these contain the number 1.



So we end up having 198+198+198=594 different variants where 2 numbers are correct and one is wrong. 33 of which have the first two in correct order. (Then, of course, there are 1-2-3 and its permutations which have 3 correct numbers and include the jackpot, so there are 6 more combinations to win FG).

It's not relevant whether you divide 561(=594-33), 594 or 600 by 42840. The result always differs (significantly) from 1:210.

PS: We can consider any variant of numbers that will get drawn. I used 1-2-3 for simplicity. The same logic applies if, let's say, 36-4-17 are picked.

This post was edited by TheOnlyDenny on Sep 21 2022 03:24am

there are 2 correct numbers in, removing them from the drawing pool entirely.

leaving 34 numbers remaining, for the third number picked. those numbers can be in any of the 6 possible combinations. 34 * 6 = 204 possible combinations with 2 numbers staying the same.

jackpot is 1-2-3 for example:
with 1-2-3 pulled
so 1-2-3. 1-3-2. 2-1-3, 2-3-1, 3-1-2, 3-2-1. that's one set of 6 for 1 of the 34 remaining numbers. and removes them from the equation for being the 3 correct number options. leaving 33 numbers with 6 potential combinations each.

Now have to remove the numbers with the other prize odds, 2 numbers ordered.

remove 6 for the 3 correct numbers.
remove 33 for the first two numbers in order.
that leaves 204-39 = 165

42,840 total combinations / 204 possible combinations with at least 2 correct numbers = 210.

6 of the possible combinations are 3 correct number options, 33 of which have the first two ordered correctly, leaving 165 possible 2 correct number combinations.

42,840/5(removing the jackpot) = 8,568. so a 1 in 8,568 chance to pull all 3 correct numbers,

42,840/33(removing all 3 correct numbers) = 1,298. So it should be 1 in 1,298, not 1 in 1,260.(1 in 1,260 is including the jackpot draw)

It's inaccurate in which it is including the 6x 3 correct number options, and 2 ordered correct numbers. it's correct in which your odds of pulling 2 correct numbers, when including the odds of a higher prize reward.

Though i believe it should actually be 42,840/165 = 1 in 259.6 chance to pull 2 correct numbers only.
If you include the higher prizes, it would be 1 in 210 as a total, as the equation on the page shows it out of the total possible combinations.


The correct odds of winning should be i believe
1 in 42,840 for jackpot
1 in 8,568 for 3 correct numbers.
1 in 1,298 for 2 correct ordered numbers.
1 in 259 or 260 if you round up for pulling 2 correct numbers.

if you include all higher prizes into the next tier, the numbers are correct though.

Edit: You pull the two numbers out of the three that get drawn (=3 combinations), then you pull one out of the 33 that won't be picked (=33), then you take into account permutations with these (so times 6). You miss the first part of that (see below):

However, I have written a small program to list any variant with 2 correct numbers if 1-2-3 are pulled. The following list has 18 lines, each of them shows 33 combinations, so we have 18*33=594. Can you spot any repetitions?

First third has no 3, second third has no 2, last on has no 1. It's obvious there's no repetition within one line. And then there are the 6 permutations, which have the same three numbers, but come in different order.


Those with no number 3 in them:


1-2-4, 1-2-5, 1-2-6, 1-2-7, 1-2-8, 1-2-9, 1-2-10, 1-2-11, 1-2-12, 1-2-13, 1-2-14, 1-2-15, 1-2-16, 1-2-17, 1-2-18, 1-2-19, 1-2-20, 1-2-21, 1-2-22, 1-2-23, 1-2-24, 1-2-25, 1-2-26, 1-2-27, 1-2-28, 1-2-29, 1-2-30, 1-2-31, 1-2-32, 1-2-33, 1-2-34, 1-2-35, 1-2-36

2-1-4, 2-1-5, 2-1-6, 2-1-7, 2-1-8, 2-1-9, 2-1-10, 2-1-11, 2-1-12, 2-1-13, 2-1-14, 2-1-15, 2-1-16, 2-1-17, 2-1-18, 2-1-19, 2-1-20, 2-1-21, 2-1-22, 2-1-23, 2-1-24, 2-1-25, 2-1-26, 2-1-27, 2-1-28, 2-1-29, 2-1-30, 2-1-31, 2-1-32, 2-1-33, 2-1-34, 2-1-35, 2-1-36

1-4-2, 1-5-2, 1-6-2, 1-7-2, 1-8-2, 1-9-2, 1-10-2, 1-11-2, 1-12-2, 1-13-2, 1-14-2, 1-15-2, 1-16-2, 1-17-2, 1-18-2, 1-19-2, 1-20-2, 1-21-2, 1-22-2, 1-23-2, 1-24-2, 1-25-2, 1-26-2, 1-27-2, 1-28-2, 1-29-2, 1-30-2, 1-31-2, 1-32-2, 1-33-2, 1-34-2, 1-35-2, 1-36-2

2-4-1, 2-5-1, 2-6-1, 2-7-1, 2-8-1, 2-9-1, 2-10-1, 2-11-1, 2-12-1, 2-13-1, 2-14-1, 2-15-1, 2-16-1, 2-17-1, 2-18-1, 2-19-1, 2-20-1, 2-21-1, 2-22-1, 2-23-1, 2-24-1, 2-25-1, 2-26-1, 2-27-1, 2-28-1, 2-29-1, 2-30-1, 2-31-1, 2-32-1, 2-33-1, 2-34-1, 2-35-1, 2-36-1

4-1-2, 5-1-2, 6-1-2, 7-1-2, 8-1-2, 9-1-2, 10-1-2, 11-1-2, 12-1-2, 13-1-2, 14-1-2, 15-1-2, 16-1-2, 17-1-2, 18-1-2, 19-1-2, 20-1-2, 21-1-2, 22-1-2, 23-1-2, 24-1-2, 25-1-2, 26-1-2, 27-1-2, 28-1-2, 29-1-2, 30-1-2, 31-1-2, 32-1-2, 33-1-2, 34-1-2, 35-1-2, 36-1-2

4-2-1, 5-2-1, 6-2-1, 7-2-1, 8-2-1, 9-2-1, 10-2-1, 11-2-1, 12-2-1, 13-2-1, 14-2-1, 15-2-1, 16-2-1, 17-2-1, 18-2-1, 19-2-1, 20-2-1, 21-2-1, 22-2-1, 23-2-1, 24-2-1, 25-2-1, 26-2-1, 27-2-1, 28-2-1, 29-2-1, 30-2-1, 31-2-1, 32-2-1, 33-2-1, 34-2-1, 35-2-1, 36-2-1


Those with no number 2 in them:


1-3-4, 1-3-5, 1-3-6, 1-3-7, 1-3-8, 1-3-9, 1-3-10, 1-3-11, 1-3-12, 1-3-13, 1-3-14, 1-3-15, 1-3-16, 1-3-17, 1-3-18, 1-3-19, 1-3-20, 1-3-21, 1-3-22, 1-3-23, 1-3-24, 1-3-25, 1-3-26, 1-3-27, 1-3-28, 1-3-29, 1-3-30, 1-3-31, 1-3-32, 1-3-33, 1-3-34, 1-3-35, 1-3-36

3-1-4, 3-1-5, 3-1-6, 3-1-7, 3-1-8, 3-1-9, 3-1-10, 3-1-11, 3-1-12, 3-1-13, 3-1-14, 3-1-15, 3-1-16, 3-1-17, 3-1-18, 3-1-19, 3-1-20, 3-1-21, 3-1-22, 3-1-23, 3-1-24, 3-1-25, 3-1-26, 3-1-27, 3-1-28, 3-1-29, 3-1-30, 3-1-31, 3-1-32, 3-1-33, 3-1-34, 3-1-35, 3-1-36

1-4-3, 1-5-3, 1-6-3, 1-7-3, 1-8-3, 1-9-3, 1-10-3, 1-11-3, 1-12-3, 1-13-3, 1-14-3, 1-15-3, 1-16-3, 1-17-3, 1-18-3, 1-19-3, 1-20-3, 1-21-3, 1-22-3, 1-23-3, 1-24-3, 1-25-3, 1-26-3, 1-27-3, 1-28-3, 1-29-3, 1-30-3, 1-31-3, 1-32-3, 1-33-3, 1-34-3, 1-35-3, 1-36-3

3-4-1, 3-5-1, 3-6-1, 3-7-1, 3-8-1, 3-9-1, 3-10-1, 3-11-1, 3-12-1, 3-13-1, 3-14-1, 3-15-1, 3-16-1, 3-17-1, 3-18-1, 3-19-1, 3-20-1, 3-21-1, 3-22-1, 3-23-1, 3-24-1, 3-25-1, 3-26-1, 3-27-1, 3-28-1, 3-29-1, 3-30-1, 3-31-1, 3-32-1, 3-33-1, 3-34-1, 3-35-1, 3-36-1

4-1-3, 5-1-3, 6-1-3, 7-1-3, 8-1-3, 9-1-3, 10-1-3, 11-1-3, 12-1-3, 13-1-3, 14-1-3, 15-1-3, 16-1-3, 17-1-3, 18-1-3, 19-1-3, 20-1-3, 21-1-3, 22-1-3, 23-1-3, 24-1-3, 25-1-3, 26-1-3, 27-1-3, 28-1-3, 29-1-3, 30-1-3, 31-1-3, 32-1-3, 33-1-3, 34-1-3, 35-1-3, 36-1-3

4-3-1, 5-3-1, 6-3-1, 7-3-1, 8-3-1, 9-3-1, 10-3-1, 11-3-1, 12-3-1, 13-3-1, 14-3-1, 15-3-1, 16-3-1, 17-3-1, 18-3-1, 19-3-1, 20-3-1, 21-3-1, 22-3-1, 23-3-1, 24-3-1, 25-3-1, 26-3-1, 27-3-1, 28-3-1, 29-3-1, 30-3-1, 31-3-1, 32-3-1, 33-3-1, 34-3-1, 35-3-1, 36-3-1


Those with no number 1 in them


2-3-4, 2-3-5, 2-3-6, 2-3-7, 2-3-8, 2-3-9, 2-3-10, 2-3-11, 2-3-12, 2-3-13, 2-3-14, 2-3-15, 2-3-16, 2-3-17, 2-3-18, 2-3-19, 2-3-20, 2-3-21, 2-3-22, 2-3-23, 2-3-24, 2-3-25, 2-3-26, 2-3-27, 2-3-28, 2-3-29, 2-3-30, 2-3-31, 2-3-32, 2-3-33, 2-3-34, 2-3-35, 2-3-36

3-2-4, 3-2-5, 3-2-6, 3-2-7, 3-2-8, 3-2-9, 3-2-10, 3-2-11, 3-2-12, 3-2-13, 3-2-14, 3-2-15, 3-2-16, 3-2-17, 3-2-18, 3-2-19, 3-2-20, 3-2-21, 3-2-22, 3-2-23, 3-2-24, 3-2-25, 3-2-26, 3-2-27, 3-2-28, 3-2-29, 3-2-30, 3-2-31, 3-2-32, 3-2-33, 3-2-34, 3-2-35, 3-2-36

2-4-3, 2-5-3, 2-6-3, 2-7-3, 2-8-3, 2-9-3, 2-10-3, 2-11-3, 2-12-3, 2-13-3, 2-14-3, 2-15-3, 2-16-3, 2-17-3, 2-18-3, 2-19-3, 2-20-3, 2-21-3, 2-22-3, 2-23-3, 2-24-3, 2-25-3, 2-26-3, 2-27-3, 2-28-3, 2-29-3, 2-30-3, 2-31-3, 2-32-3, 2-33-3, 2-34-3, 2-35-3, 2-36-3

3-4-2, 3-5-2, 3-6-2, 3-7-2, 3-8-2, 3-9-2, 3-10-2, 3-11-2, 3-12-2, 3-13-2, 3-14-2, 3-15-2, 3-16-2, 3-17-2, 3-18-2, 3-19-2, 3-20-2, 3-21-2, 3-22-2, 3-23-2, 3-24-2, 3-25-2, 3-26-2, 3-27-2, 3-28-2, 3-29-2, 3-30-2, 3-31-2, 3-32-2, 3-33-2, 3-34-2, 3-35-2, 3-36-2

4-2-3, 5-2-3, 6-2-3, 7-2-3, 8-2-3, 9-2-3, 10-2-3, 11-2-3, 12-2-3, 13-2-3, 14-2-3, 15-2-3, 16-2-3, 17-2-3, 18-2-3, 19-2-3, 20-2-3, 21-2-3, 22-2-3, 23-2-3, 24-2-3, 25-2-3, 26-2-3, 27-2-3, 28-2-3, 29-2-3, 30-2-3, 31-2-3, 32-2-3, 33-2-3, 34-2-3, 35-2-3, 36-2-3

4-3-2, 5-3-2, 6-3-2, 7-3-2, 8-3-2, 9-3-2, 10-3-2, 11-3-2, 12-3-2, 13-3-2, 14-3-2, 15-3-2, 16-3-2, 17-3-2, 18-3-2, 19-3-2, 20-3-2, 21-3-2, 22-3-2, 23-3-2, 24-3-2, 25-3-2, 26-3-2, 27-3-2, 28-3-2, 29-3-2, 30-3-2, 31-3-2, 32-3-2, 33-3-2, 34-3-2, 35-3-2, 36-3-2

This post was edited by TheOnlyDenny on Sep 21 2022 06:48am