Combinatorial Distribution

Combinatorial Distribution

    Factorial - n! = n * (n -1) * (n - 2) * ... * 3 * 2 * 1 , and 0! = 1

    Permutation - P(n,r) = n! / (n - r)!

    Combination - C(n,r) = P(n,r) / r!

    Combinatorial Distribution - D(n,r,c,z) = C(z - 1, c - 1) * C(n - z, r - c)

      n - total number of items
      r - number of items in a combinatorial or permutational set
      c - column number of the distribution
      z - item number of the distribution

Example distribution of 49 numbers taken 6 at a time (a.k.a 6/49) n = 49, r = 6:

Here is some Visual Basic code for the functions. Excel has it's own factorial, permutation, and combination functions.

Factorial VB code

Function Fact(n) As Double

    If (n < 1) Then
      Fact = 1
    Else
      Fact = n * Fact(n - 1)
    End If

End Function

Permutation VB code

Function Perm(n, r) As Double

    If (n < r) Then
      Perm = 0
    Else
      Perm = Fact(n) / Fact(n - r)
    End If

End Function

Combination VB code

Function Comb(n, r) As Double

    If (n < r) Then
      Comb = 0
    Else
      Comb = Perm(n, r) / Fact(r)
    End If

End Function

Column Distribution VB code

Function Cdist(n, r, c, z) As Double

    If (z < c) Or (z > (n - r + c)) Or (z > n) Or (c > r) Or (n < 1) Or (r < 1) Or (c < 1) Or (z < 1) Then
      Cdist = 0
    Else
      Cdist = Comb(z - 1, c - 1) * Comb(n - z, r - c)
    End If

End Function

http://www.johnph77.com/math/lf.html

Positional frequencies for all lottery matrices from 2/2 to 7/100.

http://www.johnph77.com.math/ls.html

Sum charts for all lottery matrices from 2/2 to 7/100 plus Pick 2 through Pick 6.

gl

j

Great job. Your second link is probably mistyped, but I got there anyway.

I tend to think in formulae or code and can setup tables and graphs in my mind's eye. It keeps things more compact for me. I posted the code for others to copy and paste for personal use.

TY.

It is - http://www.johnph77.com/math/ls.html.

Personally, I'm not too good with graphs and charts, but I can do tables.

And thanks for the formulas - nicely done. Although the formulas used to generate the spreadsheet that resulted in those tables were different the results are the same. Nice to have confirmation using a different approach, though. All the data in those tables was originally confirmed by programming - going through the matrices and doing an actual count of all instances with hard copy output.

gl

j

Yeah, back in about the 1988-89 time frame, I went the long way to arrive at that equation, but worth it.

Graph for 6/49, shows symmetry by color; red, green, and blue are symmetric with each other.

That is a good idea. Basically, JADE LSG does something like that when creating a wheel algorithmically; tweaked for line count(play count).

Well, there was something there... guess it was deleted.

Yes, it was deleted. It was mine, but needed further work.

The 6/49 position frequency chart as I had first developed it - it is exactly as you had published it in your original post:

...............................6/49................................
..........A........B........C........D........E........F......Total
..1....1712304..............................................1712304
..2....1533939...178365.....................................1712304
..3....1370754...326370....15180............................1712304
..4....1221759...446985....42570......990...................1712304
..5....1086008...543004....79464.....3784.......44..........1712304
..6.....962598...617050...123410.....9030......215........1.1712304
..7.....850668...671580...172200....17220......630........6.1712304
..8.....749398...708890...223860....28700.....1435.......21.1712304
..9.....658008...731120...276640....43680.....2800.......56.1712304
..10....575757...740259...329004....62244.....4914......126.1712304
..11....501942...738150...379620....84360.....7980......252.1712304
..12....435897...726495...427350...109890....12210......462.1712304
..13....376992...706860...471240...138600....17820......792.1712304
..14....324632...680680...510510...170170....25025.....1287.1712304
..15....278256...649264...544544...204204....34034.....2002.1712304
..16....237336...613800...572880...240240....45045.....3003.1712304
..17....201376...575360...595200...277760....58240.....4368.1712304
..18....169911...534905...611320...316200....73780.....6188.1712304
..19....142506...493290...621180...354960....91800.....8568.1712304
..20....118755...451269...624834...393414...112404....11628.1712304
..21.....98280...409500...622440...430920...135660....15504.1712304
..22.....80730...368550...614250...466830...161595....20349.1712304
..23.....65780...328900...600600...500500...190190....26334.1712304
..24.....53130...290950...581900...531300...221375....33649.1712304
..25.....42504...255024...558624...558624...255024....42504.1712304
..26.....33649...221375...531300...581900...290950....53130.1712304
..27.....26334...190190...500500...600600...328900....65780.1712304
..28.....20349...161595...466830...614250...368550....80730.1712304
..29.....15504...135660...430920...622440...409500....98280.1712304
..30.....11628...112404...393414...624834...451269...118755.1712304
..31......8568....91800...354960...621180...493290...142506.1712304
..32......6188....73780...316200...611320...534905...169911.1712304
..33......4368....58240...277760...595200...575360...201376.1712304
..34......3003....45045...240240...572880...613800...237336.1712304
..35......2002....34034...204204...544544...649264...278256.1712304
..36......1287....25025...170170...510510...680680...324632.1712304
..37.......792....17820...138600...471240...706860...376992.1712304
..38.......462....12210...109890...427350...726495...435897.1712304
..39.......252.....7980....84360...379620...738150...501942.1712304
..40.......126.....4914....62244...329004...740259...575757.1712304
..41........56.....2800....43680...276640...731120...658008.1712304
..42........21.....1435....28700...223860...708890...749398.1712304
..43.........6......630....17220...172200...671580...850668.1712304
..44.........1......215.....9030...123410...617050...962598.1712304
..45.................44.....3784....79464...543004..1086008.1712304
..46.........................990....42570...446985..1221759.1712304
..47................................15180...326370..1370754.1712304
..48........................................178365..1533939.1712304
..49................................................1712304.1712304
Total.13983816.13983816.13983816.13983816.13983816.13983816........

What I decided to do was to eliminate all the numbers in each column that wouldn't fall 90% of the time so as to rid my picks of more or less unreal combinations. Doing so left me with this:

..............................6/49.................................
..........A........B.........C.......D........E........F......Total
...1...1712304..............................................1712304
...2...1533939..............................................1533939
...3...1370754...326370.....................................1697124
...4...1221759...446985.....................................1668744
...5...1086008...543004.....................................1629012
...6....962598...617050.....................................1579648
...7....850668...671580.....................................1522248
...8....749398...708890...223860............................1682148
...9....658008...731120...276640............................1665768
..10....575757...740259...329004............................1645020
..11....501942...738150...379620............................1619712
..12....435897...726495...427350............................1589742
..13....376992...706860...471240............................1555092
..14....324632...680680...510510............................1515822
..15....278256...649264...544544............................1472064
..16.............613800...572880...240240...................1426920
..17.............575360...595200...277760...................1448320
..18.............534905...611320...316200...................1462425
..19.............493290...621180...354960...................1469430
..20.............451269...624834...393414...................1469517
..21.............409500...622440...430920...................1462860
..22.............368550...614250...466830...................1449630
..23.............328900...600600...500500...................1430000
..24.............290950...581900...531300...................1404150
..25.............255024...558624...558624...255024..........1627296
..26......................531300...581900...290950..........1404150
..27......................500500...600600...328900..........1430000
..28......................466830...614250...368550..........1449630
..29......................430920...622440...409500..........1462860
..30......................393414...624834...451269..........1469517
..31......................354960...621180...493290..........1469430
..32......................316200...611320...534905..........1462425
..33......................277760...595200...575360..........1448320
..34......................240240...572880...613800..........1426920
..35...............................544544...649264...278256.1472064
..36...............................510510...680680...324632.1515822
..37...............................471240...706860...376992.1555092
..38...............................427350...726495...435897.1589742
..39...............................379620...738150...501942.1619712
..40...............................329004...740259...575757.1645020
..41...............................276640...731120...658008.1665768
..42...............................223860...708890...749398.1682148
..43........................................671580...850668.1522248
..44........................................617050...962598.1579648
..45........................................543004..1086008.1629012
..46........................................446985..1221759.1668744
..47........................................326370..1370754.1697124
..48................................................1533939.1533939
..49................................................1712304.1712304
Total.12638912.12608255.12678120.12678120.12608255.12638912.8892704
........90.38%...90.16%...90.66%...90.66%...90.16%...90.38%...63.59%

It's impossible to achieve an exact 90% reduction but I came close.

If you were to reduce the six columns by an exact 90%, theoretically you would be left with 53.14% of the total possibilities. But, in fact, the remaining numbers will hold true for 63.59% of all the possible combinations, giving the bettor a 10.45% edge on the game.

As stated before, good idea. It's the same place I started when I came up with the idea for JADE LSG. I'm sure there are many other possible ways of deriving a wheel or using as a lottery number selection method.

Absolutely. And if you were to reduce them to the most common occurances in each column, you'd wind up playing 01-10-20-30-40-49 - the numbers most likely to appear in each column. But that combination has as much chance of hitting than any of the others.

gl

j

Yes, however, if you add a little Quantum Fog to the combinatorial set of {01, 10, 20, 30, 40, 49} you'll get more coverage.