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Combinatorial Distribution

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

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JADELottery's avatar - MeAtWork 03.PNG
The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3675 Posts
Offline
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.
Use at your own risk.

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

    JADELottery's avatar - MeAtWork 03.PNG
    The Quantum Master
    West Concord, MN
    United States
    Member #21
    December 7, 2001
    3675 Posts
    Offline
    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.
    Use at your own risk.

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

      johnph77's avatar - avatar
      CA
      United States
      Member #2987
      December 10, 2003
      832 Posts
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      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.

       =^.^=

        JADELottery's avatar - MeAtWork 03.PNG
        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.
        Use at your own risk.

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

          johnph77's avatar - avatar
          CA
          United States
          Member #2987
          December 10, 2003
          832 Posts
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          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.

          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

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

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

           =^.^=

            JADELottery's avatar - MeAtWork 03.PNG
            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.

            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.

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

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

              JADELottery's avatar - MeAtWork 03.PNG
              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.
              Use at your own risk.

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

                JADELottery's avatar - MeAtWork 03.PNG
                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.
                Use at your own risk.

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

                  JADELottery's avatar - MeAtWork 03.PNG
                  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.
                  Use at your own risk.

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

                    johnph77's avatar - avatar
                    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.

                     =^.^=

                      JADELottery's avatar - MeAtWork 03.PNG
                      The Quantum Master
                      West Concord, MN
                      United States
                      Member #21
                      December 7, 2001
                      3675 Posts
                      Offline
                      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.
                      Use at your own risk.

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

                        johnph77's avatar - avatar
                        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.

                         =^.^=

                          JADELottery's avatar - MeAtWork 03.PNG
                          The Quantum Master
                          West Concord, MN
                          United States
                          Member #21
                          December 7, 2001
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                          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.

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