Welcome Guest
You last visited June 26, 2017, 7:57 am
All times shown are
Eastern Time (GMT-5:00)

# Lottery Blackbody Radiation

Topic closed. 39 replies. Last post 11 months ago by Sunglasses.

 Page 1 of 3
The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3857 Posts
Online
 Posted: July 29, 2016, 6:27 pm - IP Logged

As we stated in our blog, we were asked a question by CARBOB about how long does it take for all numbers in a Pick 3 type lottery to be picked; both in a single column and overall.

It led to an interesting formula that is very similar to Blackbody Radiation.

We'll start with a simple Pick 1 example.

This can apply to any individual Pick 3 column.

In the Pick 1 selection, we keep track of the number selected at each drawing.

Below is a table showing the Draw, Picked number and the Tracked # Selection.

The example shows on the 24th drawing, all the numbers have been selected.

On the 25th draw, the cycle starts over and the Tracking begins again.

 Draw Pick 1 Tracked # Selection 0 1 2 3 4 5 6 7 8 9 1 3 · · · 1 · · · · · · 2 4 · · · 1 1 · · · · · 3 1 · 1 · 1 1 · · · · · 4 4 · 1 · 1 2 · · · · · 5 1 · 2 · 1 2 · · · · · 6 1 · 3 · 1 2 · · · · · 7 1 · 4 · 1 2 · · · · · 8 6 · 4 · 1 2 · 1 · · · 9 9 · 4 · 1 2 · 1 · · 1 10 2 · 4 1 1 2 · 1 · · 1 11 8 · 4 1 1 2 · 1 · 1 1 12 2 · 4 2 1 2 · 1 · 1 1 13 2 · 4 3 1 2 · 1 · 1 1 14 3 · 4 3 2 2 · 1 · 1 1 15 0 1 4 3 2 2 · 1 · 1 1 16 8 1 4 3 2 2 · 1 · 2 1 17 1 1 5 3 2 2 · 1 · 2 1 18 8 1 5 3 2 2 · 1 · 3 1 19 9 1 5 3 2 2 · 1 · 3 2 20 2 1 5 4 2 2 · 1 · 3 2 21 9 1 5 4 2 2 · 1 · 3 3 22 4 1 5 4 2 3 · 1 · 3 3 23 5 1 5 4 2 3 1 1 · 3 3 24 7 1 5 4 2 3 1 1 1 3 3 25 8 · · · · · · · · 1 · 26 1 · 1 · · · · · · 1 · 27 6 · 1 · · · · 1 · 1 · 28 3 · 1 · 1 · · 1 · 1 · 29 0 1 1 · 1 · · 1 · 1 · 30 1 1 2 · 1 · · 1 · 1 · 31 2 1 2 1 1 · · 1 · 1 · 32 9 1 2 1 1 · · 1 · 1 1

Continues...

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

The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3857 Posts
Online
 Posted: July 29, 2016, 6:39 pm - IP Logged

The next step was to gather a frequency of draws when the cycle ends.

In the previous example, it was the 24th draw.

We ran a simulation for Millions of draws to obtain a Million data set of end draw cycles.

Below is a table of our obtained data for this run.

The table shows there is a minimum of 10 draws; which is to be expected since there are only 10 numbers 0 to 9 that can be selected.

The Peak appears to be at 24 draws.

Next we will look at the graph of the data.

 Draws Pick 1 Tracked # Selection Frequency 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 10 344 11 1605 12 4306 13 7962 14 12670 15 19266 16 24277 17 29919 18 34560 19 38797 20 41426 21 43395 22 45105 23 44710 24 45240 25 43409 26 42265 27 40858 28 38108 29 36519 30 34598 31 31914 32 29250 33 27122 34 25506 35 23658 36 21466 37 19721 38 18198 39 16430 40 15337 41 14041 42 12429 43 11339 44 10231 45 8885 46 8500 47 7532 48 6914 49 5965 50 5333 51 4974 52 4461 53 4077 54 3847 55 3361 56 2983 57 2857 58 2536 59 2289 60 1979 61 1828 62 1573 63 1559 64 1076 65 1207 66 974 67 950 68 926 69 787 70 599 71 586 72 548 73 469 74 477 75 416 76 342 77 280 78 299 79 189 80 228 81 208 82 219 83 139 84 184 85 141 86 107 87 144 88 113 89 75 90 76 91 64 92 77 93 104 94 61 95 28 96 54 97 58 98 59 99 32 100 18 101 28 102 26 103 10 104 29 105 20 106 14 107 16 108 10 109 7 110 8 111 7 112 6 113 21 114 4 115 0 116 10 117 3 118 0 119 0 120 10 121 0 122 6 123 0 124 3 125 0 126 0 127 0 128 3 129 0 130 0 131 0 132 0 133 3 134 0 135 0 136 3 137 0 138 0 139 0 140 5

Continues ...

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

The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3857 Posts
Online
 Posted: July 29, 2016, 6:48 pm - IP Logged

In the graph of the Tracked # Selections, we can see the frequency is skewed to the low end and tapers off to the high end.

This graph looks very similar to a natural phenomenon know as Blackbody Radiation.

Below is a graph of the Planck Radiation Formula.

We tried to curve fit this formula to the data, but it did not fit very well.

We had to come up with a different variation that looks nearly the same as the Planck Formula.

Continues ...

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

The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3857 Posts
Online
 Posted: July 29, 2016, 7:20 pm - IP Logged

The formula expression we had to used is as follows.

This expression fit the data far better.

Using the expression we curve fit it and found the values for αβγ and δ

The graph next shows the bar graph and the curve fit probability function.

Continues ...

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

The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3857 Posts
Online
 Posted: July 29, 2016, 7:26 pm - IP Logged

The coefficients for the Pick 1 probability are:

 Coefficients Pick 1 α -966.93367803712 β 30.6458275075256 γ -0.165505462399517 δ -0.105317939314043

These plugged in the probability function gives the following graph.

Continues ...

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

United States
Member #139779
March 3, 2013
116 Posts
Offline
 Posted: July 29, 2016, 7:32 pm - IP Logged

The formula expression we had to used is as follows.

This expression fit the data far better.

Using the expression we curve fit it and found the values for αβγ and δ

The graph next shows the bar graph and the curve fit probability function.

Continues ...

Is this the same or similar as the wave add in you made for excel 2007?

The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3857 Posts
Online
 Posted: July 29, 2016, 7:35 pm - IP Logged

Using the coefficients and probability function we can now calculate the maximum frequency draws that it takes to draw all the numbers in a single column.

This works out to be about 22.732498 draws.

In addition, here are other points of interest in the data we sampled.

 · Draws Minimum 10 Peak 22.732498 Mode 24 Median 27 Mean 29.283008 Maximum 140

Continues ...

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

The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3857 Posts
Online
 Posted: July 29, 2016, 7:36 pm - IP Logged

Is this the same or similar as the wave add in you made for excel 2007?

This is something new.

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

United States
Member #139779
March 3, 2013
116 Posts
Offline
 Posted: July 29, 2016, 7:50 pm - IP Logged

I have been trying to follow your research. So far it has led me to test skip ranges for single digits. this does tell when to best play the number. working with the 2007 add in I found that groups of 5 worked best. I was not having much luck with straight number runs but it did work better with box digit tracking. Are you going to make a new add in for this method? I find this very interesting and appreciate your work on this. I wonder how this formula in this thread would look when applied to digit skips?

Thank you.

Belgium
Member #173932
March 26, 2016
1396 Posts
Offline
 Posted: July 29, 2016, 7:51 pm - IP Logged

Using the coefficients and probability function we can now calculate the maximum frequency draws that it takes to draw all the numbers in a single column.

This works out to be about 22.732498 draws.

In addition, here are other points of interest in the data we sampled.

 · Draws Minimum 10 Peak 22.732498 Mode 24 Median 27 Mean 29.283008 Maximum 140

Continues ...

I'd suggest to set the maximum to 140 to lifetime of the game.

The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3857 Posts
Online
 Posted: July 29, 2016, 7:52 pm - IP Logged

We expanded this tracking to all the Pick 3 numbers and were excited to find the same formula worked for the Pick 3 overall tracking.

Below is an example of the tracking.

In this example, we can see all the numbers are picked by draw 8.

Then the cycle starts over and the next complete selection is at draw 18 or 10 draws to select all numbers.

 Draw Pick 3 Tracked # Selection A B C 0 1 2 3 4 5 6 7 8 9 1 2 6 8 · · 1 · · · 1 · 1 · 2 0 1 6 1 1 1 · · · 2 · 1 · 3 6 4 0 2 1 1 · 1 · 3 · 1 · 4 0 0 8 4 1 1 · 1 · 3 · 2 · 5 5 5 6 4 1 1 · 1 2 4 · 2 · 6 1 5 1 4 3 1 · 1 3 4 · 2 · 7 9 7 2 4 3 2 · 1 3 4 1 2 1 8 0 3 4 5 3 2 1 2 3 4 1 2 1 9 0 9 2 1 · 1 · · · · · · 1 10 6 3 8 1 · 1 1 · · 1 · 1 1 11 9 0 9 2 · 1 1 · · 1 · 1 3 12 6 7 5 2 · 1 1 · 1 2 1 1 3 13 9 3 6 2 · 1 2 · 1 3 1 1 4 14 3 6 2 2 · 2 3 · 1 4 1 1 4 15 4 5 9 2 · 2 3 1 2 4 1 1 5 16 8 4 7 2 · 2 3 2 2 4 2 2 5 17 8 3 6 2 · 2 4 2 2 5 2 3 5 18 9 1 2 2 1 3 4 2 2 5 2 3 6 19 0 3 5 1 · · 1 · 1 · · · · 20 4 1 6 1 1 · 1 1 1 1 · · · 21 3 8 7 1 1 · 2 1 1 1 1 1 · 22 6 6 9 1 1 · 2 1 1 3 1 1 1 23 3 5 4 1 1 · 3 2 2 3 1 1 1 24 2 9 2 1 1 2 3 2 2 3 1 1 2 25 0 0 8 2 · · · · · · · 1 · 26 0 1 5 3 1 · · · 1 · · 1 · 27 8 3 2 3 1 1 1 · 1 · · 2 · 28 3 3 4 3 1 1 3 1 1 · · 2 · 29 8 6 0 4 1 1 3 1 1 1 · 3 · 30 3 7 0 5 1 1 4 1 1 1 1 3 · 31 2 3 3 5 1 2 6 1 1 1 1 3 · 32 6 7 7 5 1 2 6 1 1 2 3 3 ·

Continues ...

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

The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3857 Posts
Online
 Posted: July 29, 2016, 7:56 pm - IP Logged

Knowing this formula works for Pick 3, we completed it for Pick 1 to Pick 5.

Below is a table of the frequency data we sampled.

 One Million Tracked # Selections 0-9 Draws Pick 1 Pick 2 Pick 3 Pick 4 Pick 5 0 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 3995 3 0 0 0 24378 173281 4 0 0 14907 148658 301168 5 0 629 73153 220812 232686 6 0 8813 131099 199469 137767 7 0 29338 153688 147390 74468 8 0 55532 145563 98287 38103 9 0 76909 123660 62752 19275 10 344 91148 96463 38486 9677 11 1605 95825 73743 23778 4769 12 4306 93583 53976 14387 2403 13 7962 86657 38879 8607 1229 14 12670 76938 27920 5262 585 15 19266 67279 19803 3085 291 16 24277 57428 13978 1881 147 17 29919 48065 9847 1120 75 18 34560 39763 6899 659 36 19 38797 32691 5012 412 15 20 41426 26565 3379 247 15 21 43395 21883 2446 140 13 22 45105 17873 1615 82 0 23 44710 14162 1207 41 1 24 45240 11759 805 25 1 25 43409 9422 601 17 0 26 42265 7546 395 8 0 27 40858 6082 293 9 0 28 38108 4787 207 1 0 29 36519 3939 134 3 0 30 34598 3066 110 0 0 31 31914 2464 65 1 0 32 29250 1950 37 2 0 33 27122 1609 32 0 0 34 25506 1254 21 1 0 35 23658 1006 18 0 0 36 21466 854 11 0 0 37 19721 632 7 0 0 38 18198 519 5 0 0 39 16430 372 5 0 0 40 15337 335 8 0 0 41 14041 258 3 0 0 42 12429 209 1 0 0 43 11339 157 0 0 0 44 10231 126 2 0 0 45 8885 124 0 0 0 46 8500 91 1 0 0 47 7532 65 1 0 0 48 6914 50 0 0 0 49 5965 57 0 0 0 50 5333 36 0 0 0 51 4974 35 1 0 0 52 4461 25 0 0 0 53 4077 21 0 0 0 54 3847 15 0 0 0 55 3361 10 0 0 0 56 2983 14 0 0 0 57 2857 2 0 0 0 58 2536 2 0 0 0 59 2289 7 0 0 0 60 1979 2 0 0 0 61 1828 4 0 0 0 62 1573 5 0 0 0 63 1559 2 0 0 0 64 1076 0 0 0 0 65 1207 0 0 0 0 66 974 2 0 0 0 67 950 0 0 0 0 68 926 1 0 0 0 69 787 0 0 0 0 70 599 2 0 0 0 71 586 0 0 0 0 72 548 0 0 0 0 73 469 1 0 0 0 74 477 0 0 0 0 75 416 0 0 0 0 76 342 0 0 0 0 77 280 0 0 0 0 78 299 0 0 0 0 79 189 0 0 0 0 80 228 0 0 0 0 81 208 0 0 0 0 82 219 0 0 0 0 83 139 0 0 0 0 84 184 0 0 0 0 85 141 0 0 0 0 86 107 0 0 0 0 87 144 0 0 0 0 88 113 0 0 0 0 89 75 0 0 0 0 90 76 0 0 0 0 91 64 0 0 0 0 92 77 0 0 0 0 93 104 0 0 0 0 94 61 0 0 0 0 95 28 0 0 0 0 96 54 0 0 0 0 97 58 0 0 0 0 98 59 0 0 0 0 99 32 0 0 0 0 100 18 0 0 0 0 101 28 0 0 0 0 102 26 0 0 0 0 103 10 0 0 0 0 104 29 0 0 0 0 105 20 0 0 0 0 106 14 0 0 0 0 107 16 0 0 0 0 108 10 0 0 0 0 109 7 0 0 0 0 110 8 0 0 0 0 111 7 0 0 0 0 112 6 0 0 0 0 113 21 0 0 0 0 114 4 0 0 0 0 115 0 0 0 0 0 116 10 0 0 0 0 117 3 0 0 0 0 118 0 0 0 0 0 119 0 0 0 0 0 120 10 0 0 0 0 121 0 0 0 0 0 122 6 0 0 0 0 123 0 0 0 0 0 124 3 0 0 0 0 125 0 0 0 0 0 126 0 0 0 0 0 127 0 0 0 0 0 128 3 0 0 0 0 129 0 0 0 0 0 130 0 0 0 0 0 131 0 0 0 0 0 132 0 0 0 0 0 133 3 0 0 0 0 134 0 0 0 0 0 135 0 0 0 0 0 136 3 0 0 0 0 137 0 0 0 0 0 138 0 0 0 0 0 139 0 0 0 0 0 140 5 0 0 0 0

Continues ...

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

The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3857 Posts
Online
 Posted: July 29, 2016, 7:59 pm - IP Logged

The coefficients worked out to the following.

 One Million Tracked # Selections 0-9 Coefficients Pick 1 Pick 2 Pick 3 Pick 4 Pick 5 α -966.93367803712 -288.229657414759 -112.848634704551 -60.9196072763659 -45.5856415845385 β 30.6458275075256 26.4269626065139 12.9474585211997 8.68322021541142 11.5743576972091 γ -0.165505462399517 -0.00183622361668888 1.05332367290803 1.76921742869347 1.45813918513947 δ -0.105317939314043 -0.212279140684865 -0.354411199971372 -0.522830929386791 -0.679029762719017

Continues ...

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

The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3857 Posts
Online
 Posted: July 29, 2016, 8:03 pm - IP Logged

Here are the points of interest for Pick 1 to Pick 5.

 · Draws Pick 1 Pick 2 Pick 3 Pick 4 Pick 5 Minimum 10 5 4 3 2 Peak 22.732498 11.031043 7.202824 5.261479 4.031870 Mode 24 11 7 5 4 Median 27 13 8 6 5 Mean 29.283008 14.131397 9.047672 6.483503 4.933889 Maximum 140 73 51 34 24

Continues ...

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

The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3857 Posts
Online
 Posted: July 29, 2016, 8:14 pm - IP Logged

This formula is not restricted to just Daily Pick games.

We applied it to Mega Millions and Powerball.

Below is a tracked # selection data sample run for both.

 One Million Tracked # Selections Draws Mega Millions Powerball P5 MM P5 PB 0 0 0 0 0 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 5 0 0 0 0 6 0 0 0 0 7 0 0 0 0 8 0 0 0 0 9 0 0 0 0 10 0 0 0 0 11 0 0 0 0 12 0 0 0 0 13 0 0 0 0 14 0 0 0 0 15 0 6 0 0 16 0 21 0 0 17 0 53 0 0 18 0 168 0 0 19 0 470 0 0 20 0 982 0 0 21 0 1671 0 0 22 0 2511 0 0 23 0 4031 0 0 24 0 5264 0 0 25 0 7190 0 0 26 0 8643 3 0 27 0 11155 8 0 28 0 12608 11 0 29 0 15605 32 0 30 0 17408 61 0 31 8 19214 106 0 32 8 20884 245 0 33 25 22715 425 0 34 88 23963 747 0 35 134 25091 1133 12 36 285 26388 1786 0 37 366 27319 2575 36 38 718 27779 3702 24 39 1006 27975 4729 70 40 1581 28542 6435 146 41 2064 28161 7556 127 42 2869 28541 9333 321 43 3732 28112 11083 302 44 4865 27377 13536 442 45 6270 27127 15633 537 46 7434 25827 16868 1030 47 8842 25596 18930 913 48 10051 24616 21102 1338 49 12102 23820 22873 1356 50 13269 22724 23745 1891 51 15383 21957 25158 1978 52 16758 21502 26565 2566 53 18215 20001 27424 2995 54 20223 19208 27559 3253 55 20613 17998 28846 3931 56 22067 17909 28569 4051 57 23735 17046 28395 5022 58 24333 15877 28384 5282 59 24971 15429 27849 5979 60 25600 13804 26908 6655 61 26338 13253 27360 7002 62 25844 12853 26291 7782 63 26228 11885 25557 8786 64 26364 11389 24957 9426 65 25300 10759 24921 9427 66 25959 10012 23606 10125 67 25278 9633 21904 10666 68 25136 8838 22117 11952 69 24567 8398 20746 11962 70 24367 8143 19513 12203 71 23037 7349 18453 13058 72 22396 6791 18200 13282 73 21372 6413 16654 13705 74 20769 6228 16193 13050 75 19621 5888 15325 14416 76 19264 5298 14519 14426 77 17935 5123 13719 14956 78 17884 4603 12327 15210 79 16991 4450 12096 15250 80 15732 4117 10988 15151 81 15975 3858 10298 15340 82 14931 3754 9727 15700 83 14226 3311 8978 15412 84 12918 3115 8675 15023 85 12416 2772 8106 15243 86 11848 2732 7546 15816 87 11129 2532 7212 15356 88 9995 2440 6760 15153 89 9565 2329 6324 15671 90 9433 2022 5665 14222 91 8685 1988 5360 14353 92 8218 1776 4952 14211 93 7697 1704 4703 14911 94 7062 1496 4690 14660 95 6969 1548 4195 13699 96 6370 1311 3707 13853 97 5987 1413 3382 14235 98 5797 1246 3186 13164 99 5257 1212 3193 13521 100 4945 1039 2844 12957 101 4526 1019 2460 12246 102 4364 816 2358 12244 103 4094 839 2268 11867 104 3910 909 2212 12305 105 3768 730 1983 11564 106 3467 703 1842 10381 107 3026 593 1728 11491 108 3069 590 1727 11378 109 2771 609 1401 10832 110 2482 509 1390 10758 111 2252 455 1268 9948 112 1951 450 1072 9738 113 2022 439 1190 9189