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# Combinatorial Distribution

Topic closed. 12 replies. Last post 10 years ago by JADELottery.

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The Quantum Master
West Concord, MN
United States
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December 7, 2001
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 Posted: November 7, 2006, 12:38 pm - IP Logged

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:

 Column    C Item  Z ¯ 1 2 3 4 5 6 Column 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 Item Total 13983816 13983816 13983816 13983816 13983816 13983816

Presented 'AS IS' and for Entertainment Purposes Only.
Any gain or loss is your responsibility.

Order is a Subset of Chaos
Knowledge is Beyond Belief
Wisdom is Not Censored
Douglas Paul Smallish
Jehocifer

The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3675 Posts
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 Posted: November 28, 2006, 11:08 am - IP Logged

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

Presented 'AS IS' and for Entertainment Purposes Only.
Any gain or loss is your responsibility.

Order is a Subset of Chaos
Knowledge is Beyond Belief
Wisdom is Not Censored
Douglas Paul Smallish
Jehocifer

CA
United States
Member #2987
December 10, 2003
832 Posts
Offline
 Posted: November 29, 2006, 12:42 am - IP Logged

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

Blessed Saint Leibowitz, keep 'em dreamin' down there.....

Next week's convention for Psychics and Prognosticators has been cancelled due to unforeseen circumstances.

=^.^=

The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3675 Posts
Offline
 Posted: November 29, 2006, 1:13 am - IP Logged

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.

Presented 'AS IS' and for Entertainment Purposes Only.
Any gain or loss is your responsibility.

Order is a Subset of Chaos
Knowledge is Beyond Belief
Wisdom is Not Censored
Douglas Paul Smallish
Jehocifer

CA
United States
Member #2987
December 10, 2003
832 Posts
Offline
 Posted: November 29, 2006, 7:57 am - IP Logged

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.

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

Blessed Saint Leibowitz, keep 'em dreamin' down there.....

Next week's convention for Psychics and Prognosticators has been cancelled due to unforeseen circumstances.

=^.^=

The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3675 Posts
Offline
 Posted: November 29, 2006, 8:00 am - IP Logged

TY.

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.

Presented 'AS IS' and for Entertainment Purposes Only.
Any gain or loss is your responsibility.

Order is a Subset of Chaos
Knowledge is Beyond Belief
Wisdom is Not Censored
Douglas Paul Smallish
Jehocifer

The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3675 Posts
Offline
 Posted: December 1, 2006, 10:35 am - IP Logged

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

Presented 'AS IS' and for Entertainment Purposes Only.
Any gain or loss is your responsibility.

Order is a Subset of Chaos
Knowledge is Beyond Belief
Wisdom is Not Censored
Douglas Paul Smallish
Jehocifer

The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3675 Posts
Offline
 Posted: December 1, 2006, 7:43 pm - IP Logged

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

Presented 'AS IS' and for Entertainment Purposes Only.
Any gain or loss is your responsibility.

Order is a Subset of Chaos
Knowledge is Beyond Belief
Wisdom is Not Censored
Douglas Paul Smallish
Jehocifer

The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3675 Posts
Offline
 Posted: December 1, 2006, 7:44 pm - IP Logged

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.

Presented 'AS IS' and for Entertainment Purposes Only.
Any gain or loss is your responsibility.

Order is a Subset of Chaos
Knowledge is Beyond Belief
Wisdom is Not Censored
Douglas Paul Smallish
Jehocifer

CA
United States
Member #2987
December 10, 2003
832 Posts
Offline
 Posted: December 1, 2006, 10:34 pm - IP Logged

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.

Blessed Saint Leibowitz, keep 'em dreamin' down there.....

Next week's convention for Psychics and Prognosticators has been cancelled due to unforeseen circumstances.

=^.^=

The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3675 Posts
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 Posted: December 2, 2006, 3:07 am - IP Logged

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.

Presented 'AS IS' and for Entertainment Purposes Only.
Any gain or loss is your responsibility.

Order is a Subset of Chaos
Knowledge is Beyond Belief
Wisdom is Not Censored
Douglas Paul Smallish
Jehocifer

CA
United States
Member #2987
December 10, 2003
832 Posts
Offline
 Posted: December 2, 2006, 4:02 am - IP Logged

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

Blessed Saint Leibowitz, keep 'em dreamin' down there.....

Next week's convention for Psychics and Prognosticators has been cancelled due to unforeseen circumstances.

=^.^=

The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3675 Posts
Offline
 Posted: December 2, 2006, 4:07 am - IP Logged

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.

Presented 'AS IS' and for Entertainment Purposes Only.
Any gain or loss is your responsibility.