I wrote:
"Martin-888" wrote:
It is part of a lottery strategy, but it does imply buying
all the 320 combinations [...], therefore guaranteeing at
least one win of the 4 out of 6 prize.
Sounds like a scam.
Please provide a URL (http://...) for the website that describes
the strategy or offers the product.
Aha! I believe you are talking about what is called "lottery wheels" and
"lottery wheeling" in English.
But I believe you might have misunderstood some details. And quite
understandably so, especially if you are reading English descriptions, and
English is not your native language, as you indicated. I found it difficult
to understand some of the details, and English __is__ my native language
:-). I finally "got it" after reading several descriptions and examples.
I wrote:
For now, I stand by my finding that you would have to buy 2240
tickets in order to be sure to match at least 4 of 6.
I am even more confident now that I correctly "wheeled" the full set of 36
numbers to guarantee "a minimum 4-number match", as the "wheelers" put it.
I will be double-checking my algorithm later, just to be sure I don't have a
defect.
Nevertheless, I suspect that is not exactly what you wanted, at least not
according to one detail that you posted.
I wrote:
"Martin-888" wrote:
I'm not very good in math, but since there is 1 chance
out of 320, it made me believe that there was many groups
of 320 combinations.
[....]
I still do not understand where 320 comes from.
I believe I do, now. But I wonder if you really meant to write 1 chance in
389, 325, or 315, not 320.
There are two common flavors of lottery wheeling: full wheel and
abbreviated wheel. (There are other flavors, as well.)
With a full wheel, you typically choose a subset of the 36 numbers, say 18
numbers. Then we generate __all__ combinations of 6 of the __18__ numbers,
not 36, with unique sets of 4. That would be a total of 42 combinations in
this case.
With an abbreviated wheel, again you choose a subset of the 36 numbers, say
18 numbers. Then we generate a __subset__ of the combinations 6 of the
__18__ numbers with unique set of 4. That would be fewer than 42
combinations in this case. (See below for how we define the subset of
qualified combinations.)
The key difference is the condition under which the wheel "guarantees" a
minimum 4-number match.
With a full wheel, a minimum 4-number match is "guaranteed" as long as
__all_6__ of the drawn numbers (by the lottery) are in your subset of 18
numbers.
With an abbreviate wheel, a minimum 4-number match is "guaranteed" as long
as, for example, at least __4__ of the 6 drawn numbers are in your subset of
18 numbers.
At issue is the word "guaranteed". It is really a __conditional__
guarantee.
Given the condition for full wheels ("all 6 of the drawn numbers are in your
subset"), the probability that the condition is met is COMBIN(18,6) /
COMBIN(36,6) for a subset is 18 numbers.
Similarly, for abbreviated wheels, the probability that the condition ("at
least 4 of the drawn numbers are in your subset") is met is COMBIN(18,4) /
COMBIN(36/6).
This is where the "1 chance in 320" comes from: ostensibly, COMBIN(k,6) /
COMBIN(36,6) is 1/320 for some subset of k numbers.
However, I am unable to find any k for which the conditional probability is
exactly 1/320.
It would be 1/389 (1 chance in 389) for a full wheel with a subset of 15
numbers.
It would be 1/325 for an abbreviated wheel with a subset of 21 numbers and
the condition that at least 4 of the 6 drawn numbers are in the subset.
And it would be 1/315 for an abbreviated wheel with a subset of 17 numbers
and the condition that at least 5 of the 6 drawn numbers are in the subset.
Does any of that sound familiar -- closer to the facts in your circumstance?
I wrote:
Of course, that makes sense to do only if the payout
for matching exactly 4 is more than $2240, assuming $1
per ticket.
I neglected to also take into consideration other possible lesser matches
with some of the other combinations. That is too complicated to explain
further. I hope you can imagine what I mean.
PS: It might useful if I modify my algorithm to handle any full and
abbreviated wheel characteristics. Something for the future.