"Martin-888" wrote:
I think joeu is right, it is 2240 tickets that I would need
to make sure that I win 4 out of 6
Actually, the correct number is 5456. For details, see my response to
PeterT at 1:36 PM on July 15 Pacific Time.
In a nutshell, I believe we are discussing two different problems. Problem
#1: find all combinations with unique sets of 4-number subsets ("quads").
Problem #2: find the minimum number of combinations that guarantees
matching 4 of 6 drawn numbers.
I think we both assumed that a solution to Problem #1 is also a solution to
Problem #2.
Now I realize that is incorrect. In order to generate the minimum number of
combinations that truly guarantees a match of 4 out of 6 drawn numbers, we
must allow some combinations to have duplicate quads. Again, see my
response to PeterT for details.
"Martin-888" wrote:
Here is the link of the game:
http://tinyurl.com/7pvqqwe
They may have rounded the chances of win to 320 from a
different number... since you guys cannot get to 320.
Oh, I can compute __that__ probability very easily.
The important thing is: that probability is very different from the
conditional probability for full and abbreviated wheels, which I had thought
your were referring to.
The probability 1/320 has nothing to do with the minimum number of
combinations that guarantees matching 4 of 6 drawn numbers.
Moreover, when you see probabilities like 1/320, it usually does not mean
that we have counted only 320 of something. Instead, it is usually the
result of dividing one number by another.
In this case, there are 6090 ways to match 4 of 6 drawn numbers, and there
are 1,947,792 total combinations of 6 of 36 numbers, what the player
chooses.
So the probablility is 6090/1947792. Equivalently, the denominator of the
odds ("1 out of ...") is 1947792/6090, which is about 319.83. That's where
1/320 comes from.
(The obvious rounding was never an issue.)
-----
For the benefit of anyone who might want some mathematical details, the key
to counting 6090 lies in the procedure for this particular lottery, Canada's
"Jour de Paye".
The lottery (not the player) actually draws 7 of 36 numbers, not just 6 of
36. The 7th number is designated as a "bonus" number. When the lottery
speaks of matching "4 of 6", they mean 4 of the first 6 drawn numbers. When
they speak of matching "4 of 6 plus bonus", they mean that the player's 6
numbers includes 4 of the first 6 drawn numbers and the 7th bonus number.
(Note: This is very different from the powerball-type lotteries that I am
familiar with, where the bonus ball is drawn from a separate set of numbers,
and the player chooses a bonus number in addition to the "normal" numbers,
just as the lottery does.)
Consequently, the number of ways to match 4 of 6 drawn numbers (and not the
bonus) is COMBIN(6,4)*COMBIN(29,2).
That is: the number of ways to match 4 of 6 drawn numbers, times the number
of ways to choose 2 more numbers (for a total of 6) from the set of numbers
that does not include any of the 7 drawn numbers (29 = 36 - 7).
Note: The count 6525 that I posted before was calculated by
COMBIN(6,4)*COMBIN(30,2). That would be correct if the lottery drew only 6
balls. I was unaware of the "Jour de Paye" procedures until now, when
Martin finally provided the link to the game.