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
"
For the Mega Millions current matrix of 5 of 75 the distribution is as follows:
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
"
For the Mega Millions current matrix of 5 of 75 the distribution is as follows:
Column C
Item Z
1
2
3
4
5
1
1150626
2
1088430
62196
3
1028790
119280
2556
4
971635
171465
7455
71
5
916895
218960
14490
280
1
6
864501
261970
23460
690
5
7
814385
300696
34170
1360
15
8
766480
335335
46431
2345
35
9
720720
366080
60060
3696
70
10
677040
393120
74880
5460
126
11
635376
416640
90720
7680
210
12
595665
436821
107415
10395
330
13
557845
453840
124806
13640
495
14
521855
467870
142740
17446
715
15
487635
479080
161070
21840
1001
16
455126
487635
179655
26845
1365
17
424270
493696
198360
32480
1820
18
395010
497420
217056
38760
2380
19
367290
498960
235620
45696
3060
20
341055
498465
253935
53295
3876
21
316251
496080
271890
61560
4845
22
292825
491946
289380
70490
5985
23
270725
486200
306306
80080
7315
24
249900
478975
322575
90321
8855
25
230300
470400
338100
101200
10626
26
211876
460600
352800
112700
12650
27
194580
449696
366600
124800
14950
28
178365
437805
379431
137475
17550
29
163185
425040
391230
150696
20475
30
148995
411510
401940
164430
23751
31
135751
397320
411510
178640
27405
32
123410
382571
419895
193285
31465
33
111930
367360
427056
208320
35960
34
101270
351780
432960
223696
40920
35
91390
335920
437580
239360
46376
36
82251
319865
440895
255255
52360
37
73815
303696
442890
271320
58905
38
66045
287490
443556
287490
66045
39
58905
271320
442890
303696
73815
40
52360
255255
440895
319865
82251
41
46376
239360
437580
335920
91390
42
40920
223696
432960
351780
101270
43
35960
208320
427056
367360
111930
44
31465
193285
419895
382571
123410
45
27405
178640
411510
397320
135751
46
23751
164430
401940
411510
148995
47
20475
150696
391230
425040
163185
48
17550
137475
379431
437805
178365
49
14950
124800
366600
449696
194580
50
12650
112700
352800
460600
211876
51
10626
101200
338100
470400
230300
52
8855
90321
322575
478975
249900
53
7315
80080
306306
486200
270725
54
5985
70490
289380
491946
292825
55
4845
61560
271890
496080
316251
56
3876
53295
253935
498465
341055
57
3060
45696
235620
498960
367290
58
2380
38760
217056
497420
395010
59
1820
32480
198360
493696
424270
60
1365
26845
179655
487635
455126
61
1001
21840
161070
479080
487635
62
715
17446
142740
467870
521855
63
495
13640
124806
453840
557845
64
330
10395
107415
436821
595665
65
210
7680
90720
416640
635376
66
126
5460
74880
393120
677040
67
70
3696
60060
366080
720720
68
35
2345
46431
335335
766480
69
15
1360
34170
300696
814385
70
5
690
23460
261970
864501
71
1
280
14490
218960
916895
72
71
7455
171465
971635
73
2556
119280
1028790
74
62196
1088430
75
1150626
Next, if we sum the frequencies in the first column for the range 01 to 12 we get:
Now, this tells the frequency when that range hits, however, it's when it does not hit when the consecutive run ends.
So, we need the frequency of when that range does not hit; we simply subtract it from the total possible outcomes of 17259390 to get the correct frequency we need to figure out the reoccurrence distribution.
The frequency is 17259390 - 10230543 = 7028847
From Discharging Reoccurrence Distribution
" Discharge Reoccurrence - y = (d / m2 ) e-(x / m )
d - total number of draws
m - average rate of reoccurrence
x - draw difference or Dd between two draw occurrences of the same number
y - the approximate frequency of reoccurrence for a given draw difference or Dd for a given draw count d
"
m = 17259390 / 7028847 and reduces to 1917710 / 780983 which is approximately 2.45550799441
We can now use the Discharge Reoccurrence formula to see approximately what kind of distribution we might see for the current 44 draws in the mega millions pick 5 of 75.
y = (44 / (2.45550799441)2 ) e-(x / 2.45550799441 )
x - Consecutive Run of 01 to 12 Drawn
y - Rounded Approximation
Actual Observation
Percent Probability
1
5
3
11.04%
2
3
4
7.34%
3
2
0
4.89%
4
1
2
3.25%
5
1
0
2.16%
6
1
0
1.44%
7
0
0
0.96%
8
0
1
0.64%
9
0
0
0.42%
10
0
0
0.28%
11
0
0
0.19%
12
0
0
0.13%
13
0
0
0.08%
14
0
0
0.06%
15
0
0
0.04%
16
0
0
0.02%
17
0
0
0.02%
18
0
0
0.01%
The percent probability is just the e-(x / m ) portion of the formula.
To get other draw distributions, just change d to some other value.
Based on what we can determine from this, the selection from 01 to 12 is very likely to end for this consecutive run.
Now, this tells the frequency when that range hits, however, it's when it does not hit when the consecutive run ends.
So, we need the frequency of when that range does not hit; we simply subtract it from the total possible outcomes of 17259390 to get the correct frequency we need to figure out the reoccurrence distribution.
The frequency is 17259390 - 10230543 = 7028847
From Discharging Reoccurrence Distribution
" Discharge Reoccurrence - y = (d / m2 ) e-(x / m )
d - total number of draws
m - average rate of reoccurrence
x - draw difference or Dd between two draw occurrences of the same number
y - the approximate frequency of reoccurrence for a given draw difference or Dd for a given draw count d
"
m = 17259390 / 7028847 and reduces to 1917710 / 780983 which is approximately 2.45550799441
We can now use the Discharge Reoccurrence formula to see approximately what kind of distribution we might see for the current 44 draws in the mega millions pick 5 of 75.
y = (44 / (2.45550799441)2 ) e-(x / 2.45550799441 )
x - Consecutive Run of 01 to 12 Drawn
y - Rounded Approximation
Actual Observation
Percent Probability
1
5
3
11.04%
2
3
4
7.34%
3
2
0
4.89%
4
1
2
3.25%
5
1
0
2.16%
6
1
0
1.44%
7
0
0
0.96%
8
0
1
0.64%
9
0
0
0.42%
10
0
0
0.28%
11
0
0
0.19%
12
0
0
0.13%
13
0
0
0.08%
14
0
0
0.06%
15
0
0
0.04%
16
0
0
0.02%
17
0
0
0.02%
18
0
0
0.01%
The percent probability is just the e-(x / m ) portion of the formula.
To get other draw distributions, just change d to some other value.
Based on what we can determine from this, the selection from 01 to 12 is very likely to end for this consecutive run.
- Correction -
We said, "The percent probability is just the e-(x / m ) portion of the formula."
The correct probability is (1 / (2.45550799441)2 ) e-(x / 2.45550799441 ).
The (1 / (2.45550799441)2 ) propotion is required.
The proportion can also be expressed as (1917710 / 780983)2.