Theoritical Minimum Bound of Coverings (Wheels)

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lottoarchitect
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Theoritical Minimum Bound of Coverings (Wheels)

Post by lottoarchitect » Sun Jan 27, 2008 7:43 am

TRANSFERED FROM OLD FORUM

This is a technical article on the general lowest bound of a wheel. I put it here as an additional bit of information about wheels.

The general lowest bound L(n,k,t,m)=b where
n=total balls played
k=block size (6 ball lotteries k=6)
t=the guarantee we want
m=match condition
b=total minimum tickets possible (absolute bound-not necessarily match the real possible bound)

is defined as

b=e/f

where e=nCk(n,m) [ combination formula nCk(n,k)=n! / (k! * (n-k)!) ]

and f=S nCk(k,i)*nCk(n-k,m-i)

where S is the Sum mathematical symbol for i=t up to i=m

This equation comes from a Kari J. Nurmela and Patric R. J. Ostergard technical paper.

The e above defines the total combinations that have to be covered to ensure 100% guarantee and the f defines the total combinations (maximum possible) that can be covered by a single ticket in your wheel.

cheers
lottoarchitect

[HR][/HR]

Hi LA,

So, this means that the minimum possible in some cases is not true even in the future development of wheels?,

It means that the minimum possible is a higher value than the result of the formula?

Regards

[HR][/HR]

Hi Hyperdimensional,

In many many cases this is true; the formula will indicate a lower bound that is impossible to reach. There are other specific formulas (proven of course and much more complicated) for very limited situations that show a higher bound which is the real minimum (mostly on t=m categories and for certain classes of number size). Unfortunately they cover a very small area of the wheels' lowest bound problem. As soon as two tickets overlap in some of their covered combinations, we cannot reach the minimum defined by the equation. This unavoidable overlap occurs in many t<m cases.
But we can be assured that no matter what we try, there is no way to go below what the above formula produces, for any type of wheel.

kepler
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Re: Theoritical Minimum Bound of Coverings (Wheels)

Post by kepler » Tue Jun 23, 2009 12:06 pm

Hi,

I think I've submited a post that the answer is here. But I have some doubts regarding the notations:

1) Does nCk(n,m) is the same as C(n,m) - number of combinations of n, m at m?
2) What is min(k,m)?
3) Is, so, f= S (C(k,i)*C(n-k,m-i)) ?

Kind regards,

Kepler

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Re: Theoritical Minimum Bound of Coverings (Wheels)

Post by lottoarchitect » Tue Jun 23, 2009 4:31 pm

Hi Kepler,

to your questions:
1) Yes it is the same. I avoid naming it as c(...) because this c notation is used in covering designs.
2)Min(x,y) means we pick the minimum between two values x & y.
3)With the c notation, yes this is the equation.

cheers
lottoarchitect

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