Lottery Analysis: Probability Theory vs. 12,954 Actual Draws

So why analyze thousands of draws at all? Because lottery analysis can prove one thing. Since 2017, I have been computing the theoretical long-run frequency of every combinatorial group in a lottery game. I wanted to know how closely actual draws follow those computations. So I compared the probability expectations against 12,954 real draws from popular US lottery games, including Powerball, Mega Millions, and Lotto America.

The short answer: actual draws track the math. The combinatorial groups that occupy a bigger share of all possible combinations show up more often, in roughly the proportion probability theory says they should. And the longer a game runs, the tighter the agreement gets. That is the law of large numbers doing exactly what it is supposed to do.

The full data tables for each game are below.

Let me get the important part out of the way. No amount of lottery analysis can predict the next draw. Not mine, not anyone’s. Every draw is an independent random event, and the balls have no memory.

Table of Contents

What lottery statistics can and cannot tell you

Most lottery statistics you will find online are frequency charts: which numbers came up most often, which ones are “hot,” which ones are “overdue.” The unspoken promise behind these charts is that past frequency hints at future draws.

It does not. A number that appeared ten times last month has exactly the same probability of appearing in the next draw as a number that has not shown up all year. Each drawing is independent. Nothing carries over.

That does not make statistics useless here. It changes the question. Instead of asking “what will come up next?” (which has no answer), I ask “do actual draws behave the way probability theory computes they should?” (which does). The first question is fortune telling. The second is verification, and verification is what this page is about.

How the analysis works

Every game I study gets the same treatment. I divide the game’s number field into four sets: low-odd, low-even, high-odd, and high-even. Every possible combination can then be classified by its combinatorial composition, meaning how many of its numbers come from each set. In my framework, each distinct composition is called a Lotterycodex template.

Here is why that classification matters. I can count exactly how many combinations belong to each template. From that count, I can compute each template’s probability, and from the probability, its expected long-run frequency. Then I put the expectation next to the observed frequency in the actual draw history and see how the two compare.

If a lottery is truly random, observed and expected frequencies should converge as draws pile up. If they drifted apart instead, something would be wrong, either with the theory or with the drawings. Across all four games below, they converge.

Powerball statistical analysis

US Powerball 5/69, draws from January 1, 2020 to February 9, 2026 (865 draws)

In Powerball, a small number of combinatorial composition groups account for a large share of all possible combinations. Those groups appear more often over time not because of any special property, but because they carry more combinatorial weight. That is the law of large numbers, nothing more.

Expected frequencies come from combinatorial probability; observed frequencies come from the actual draw history.

Lotto Name:	US Powerball
Date range:	January 1, 2020 to February 9, 2026
Total draws:	865 draws
	Theoretical Expected Frequency
	Observed frequency from US Powerball's actual lottery draws

Template	Expected Frequency vs Actual Frequency
#1	58 57
#2	54 62
#3	54 52
#4	54 54
#5	27 25
#6	27 24
#7	27 39
#8	27 25
#9	27 33
#10	27 28
#11	26 24
#12	26 23
#13	26 22
#14	24 28
#15	24 21
#16	24 22
#17	18 14
#18	18 20
#19	18 18
#20	16 11
#21	16 10
#22	16 17
#23	16 23
#24	16 21
#25	16 17
#26	15 14
#27	15 12
#28	15 15
#29	9 11
#30	9 6
#31	9 8
#32	8 8
#33	8 8
#34	8 4
#35	7 5
#36	7 3
#37	7 15
#38	7 13
#39	7 7
#40	7 5
#41	4 1
#42	4 5
#43	4 5
#44	3 3
#45	3 4
#46	3 3
#47	3 2
#48	3 3
#49	3 1
#50	3 4
#51	3 3
#52	3 6
#53	1 0
#54	~0 0
#55	~0 1
#56	~0 0

Mega Millions statistical analysis

Mega Millions, draws from January 2, 2018 to February 6, 2026 (834 draws)

Mega Millions has 56 distinct templates. Four of them carry the highest long-run relative frequency based on probability theory. The table shows how the actual 834 draws compare against those expectations.

Lotto Name:	Mega Millions
Date range:	January 2, 2018 to February 6, 2026
Total draws:	834 draws
	Theoretical Expected Frequency
	Observed frequency from Mega Millions's actual lottery draws

Template	Expected Frequency vs Actual Frequency
#1	55 66
#2	55 59
#3	52 62
#4	52 49
#5	27 39
#6	27 20
#7	26 23
#8	26 25
#9	26 26
#10	26 19
#11	24 26
#12	24 12
#13	24 24
#14	24 25
#15	23 22
#16	23 20
#17	17 21
#18	17 20
#19	17 14
#20	17 6
#21	16 17
#22	16 23
#23	15 16
#24	15 17
#25	14 17
#26	14 7
#27	14 13
#28	14 17
#29	9 8
#30	9 6
#31	8 11
#32	8 5
#33	8 15
#34	8 5
#35	7 3
#36	7 8
#37	7 12
#38	7 4
#39	6 4
#40	6 2
#41	4 7
#42	4 5
#43	4 7
#44	4 4
#45	4 3
#46	4 2
#47	3 6
#48	3 1
#49	3 3
#50	3 2
#51	3 2
#52	3 2
#53	1 1
#54	1 1
#55	~0 0
#56	~0 0

Lotto America statistical analysis

Lotto America, draws from December 27, 2017 to February 9, 2026 (987 draws)

The table below compares combinatorial templates by long-run frequency under probability theory, covering nearly the entire modern run of the game since its 2017 relaunch. Compositions with more combinations behind them are expected to appear more often, and 987 draws is enough history to see how well that expectation holds.

Lotto Name:	Lotto America
Date range:	December 27, 2017 to February 9, 2026
Total draws:	987 draws
	Theoretical Expected Frequency
	Observed frequency from Lotto America's actual lottery draws

Template	Expected Frequency vs Actual Frequency
#1	65 57
#2	65 78
#3	65 69
#4	65 69
#5	30 30
#6	30 34
#7	30 25
#8	30 30
#9	30 40
#10	30 22
#11	30 28
#12	30 27
#13	30 32
#14	30 28
#15	30 22
#16	30 27
#17	18 12
#18	18 28
#19	18 18
#20	18 22
#21	18 28
#22	18 25
#23	18 24
#24	18 21
#25	18 17
#26	18 14
#27	18 11
#28	18 27
#29	8 8
#30	8 6
#31	8 8
#32	8 5
#33	8 11
#34	8 4
#35	8 7
#36	8 7
#37	8 7
#38	8 8
#39	8 8
#40	8 3
#41	4 3
#42	4 2
#43	4 6
#44	4 6
#45	4 2
#46	4 1
#47	4 2
#48	4 1
#49	4 2
#50	4 4
#51	4 6
#52	4 4
#53	~0 0
#54	~0 0
#55	~0 0
#56	~0 1

More lottery games

I run the same comparison for smaller state and regional games. The datasets vary in size, but the story does not change: observed frequencies follow combinatorial probability.

Cash4Life ended in February 2026, when participating lotteries replaced it with the new Millionaire for Life game. That actually makes this dataset more interesting, not less. It is now a closed sample of 2,589 draws, and a closed sample this large is about as good as it gets for checking how well theory holds up against reality.

This is the largest dataset in the study, covering almost eleven years of drawings. In Cash4Life, a few combinatorial composition groups dominate the draw history.

Lotto Name:	Cash4Life
Date range:	May 11, 2015 to February 9, 2026
Total draws:	2,589 draws
	Theoretical Expected Frequency
	Observed frequency from Cash4Life's actual lottery draws

Template	Expected Frequency vs Actual Frequency
#1	168 168
#2	168 178
#3	168 150
#4	168 160
#5	78 96
#6	78 88
#7	78 73
#8	78 83
#9	78 77
#10	78 78
#11	78 76
#12	78 67
#13	78 96
#14	78 76
#15	78 94
#16	78 62
#17	49 53
#18	49 51
#19	49 56
#20	49 44
#21	49 43
#22	49 41
#23	49 45
#24	49 42
#25	49 42
#26	49 39
#27	49 55
#28	49 46
#29	23 27
#30	23 22
#31	23 20
#32	23 18
#33	23 30
#34	23 19
#35	23 24
#36	23 28
#37	23 17
#38	23 31
#39	23 24
#40	23 24
#41	10 5
#42	10 10
#43	10 12
#44	10 14
#45	10 9
#46	10 9
#47	10 9
#48	10 7
#49	10 7
#50	10 11
#51	10 11
#52	10 13
#53	1 1
#54	1 3
#55	1 3
#56	1 2

Lotto Name:	Arizona Fantasy 5
Date range:	January 1, 2023 to February 9, 2026
Total draws:	1,110 draws
	Theoretical Expected Frequency
	Observed frequency from Arizona Fantasy 5's actual lottery draws

Template	Expected Frequency vs Actual Frequency
#1	81 75
#2	73 70
#3	73 73
#4	73 69
#5	37 41
#6	37 42
#7	37 41
#8	37 27
#9	37 38
#10	37 40
#11	33 29
#12	33 25
#13	33 37
#14	30 33
#15	30 37
#16	30 32
#17	24 24
#18	24 37
#19	24 17
#20	20 17
#21	20 21
#22	20 21
#23	20 17
#24	20 27
#25	20 26
#26	18 19
#27	18 18
#28	18 14
#29	11 13
#30	11 8
#31	11 10
#32	10 4
#33	10 6
#34	10 11
#35	8 5
#36	8 9
#37	8 13
#38	8 8
#39	8 6
#40	8 6
#41	5 3
#42	5 6
#43	5 3
#44	3 6
#45	3 1
#46	3 1
#47	3 3
#48	3 5
#49	3 3
#50	3 4
#51	3 3
#52	3 4
#53	1 1
#54	~0 1
#55	~0 0
#56	~0 0

Lucky for Life and Cash4Life are both gone. The two multi-state draw games were retired together to make room for Millionaire for Life, a single game replacing both. Lucky for Life held its final drawing on February 21, 2026, which means its entire draw history is now closed and fixed. A sample that large and that final is about as good as it gets for checking how well probability expectations hold up against actual results.

Lotto Name:	Lucky for Life
Date range:	May 21, 2015 to February 9, 2026
Total draws:	2,094 draws
	Theoretical Expected Frequency
	Observed frequency from Lucky for Life's actual lottery draws

Template	Expected Frequency vs Actual Frequency
#1	139 177
#2	139 146
#3	139 145
#4	139 130
#5	64 64
#6	64 64
#7	64 51
#8	64 68
#9	64 58
#10	64 65
#11	64 57
#12	64 53
#13	64 77
#14	64 72
#15	64 56
#16	64 53
#17	39 31
#18	39 58
#19	39 39
#20	39 40
#21	39 33
#22	39 41
#23	39 38
#24	39 44
#25	39 38
#26	39 42
#27	39 32
#28	39 39
#29	18 16
#30	18 14
#31	18 23
#32	18 17
#33	18 19
#34	18 16
#35	18 12
#36	18 22
#37	18 13
#38	18 17
#39	18 17
#40	18 18
#41	7 2
#42	7 5
#43	7 9
#44	7 6
#45	7 9
#46	7 7
#47	7 12
#48	7 6
#49	7 5
#50	7 8
#51	7 4
#52	7 3
#53	1 0
#54	1 1
#55	1 0
#56	1 2

Lotto Name:	Gimme5
Date range:	November 18, 2016 to February 9, 2026
Total draws:	1,914 draws
	Theoretical Expected Frequency
	Observed frequency from Gimme5's actual lottery draws

Template	Expected Frequency vs Actual Frequency
#1	135 121
#2	135 142
#3	135 150
#4	120 108
#5	67 85
#6	67 49
#7	67 60
#8	61 51
#9	61 52
#10	61 62
#11	54 58
#12	54 47
#13	54 56
#14	54 59
#15	54 61
#16	54 61
#17	40 23
#18	40 35
#19	40 35
#20	36 43
#21	36 37
#22	36 44
#23	36 30
#24	36 37
#25	36 31
#26	28 29
#27	28 25
#28	28 29
#29	18 19
#30	18 20
#31	18 26
#32	18 15
#33	18 23
#34	18 15
#35	14 13
#36	14 21
#37	14 16
#38	13 11
#39	13 16
#40	13 8
#41	7 5
#42	7 8
#43	7 11
#44	7 6
#45	7 6
#46	7 11
#47	6 7
#48	6 9
#49	6 7
#50	4 3
#51	4 8
#52	4 7
#53	1 2
#54	1 1
#55	1 0
#56	~0 0

Lotto Name:	Tri-State Megabucks
Date range:	November 16, 2016 to February 9, 2026
Total draws:	1,083 draws
	Theoretical Expected Frequency
	Observed frequency from Tri-State Megabucks's actual lottery draws

Template	Expected Frequency vs Actual Frequency
#1	79 80
#2	72 50
#3	72 77
#4	72 66
#5	36 44
#6	36 39
#7	36 37
#8	36 42
#9	36 47
#10	36 32
#11	32 35
#12	32 35
#13	32 27
#14	29 32
#15	29 37
#16	29 22
#17	24 34
#18	24 17
#19	24 26
#20	19 18
#21	19 22
#22	19 24
#23	19 18
#24	19 15
#25	19 21
#26	17 15
#27	17 15
#28	17 16
#29	11 8
#30	11 11
#31	11 8
#32	10 9
#33	10 8
#34	10 6
#35	8 5
#36	8 11
#37	8 10
#38	8 6
#39	8 2
#40	8 7
#41	5 6
#42	5 6
#43	5 5
#44	3 4
#45	3 3
#46	3 2
#47	3 1
#48	3 5
#49	3 3
#50	3 1
#51	3 6
#52	3 5
#53	1 1
#54	~0 0
#55	~0 1
#56	~0 0

Lotto Name:	Arkansas Lotto
Date range:	September 21, 2022 to February 7, 2026
Total draws:	354 draws
	Theoretical Expected Frequency
	Observed frequency from Arkansas Lotto's actual lottery draws

Template	Expected Frequency vs Actual Frequency
#1	19 14
#2	19 23
#3	19 14
#4	19 13
#5	19 17
#6	19 19
#7	11 10
#8	11 13
#9	11 11
#10	11 12
#11	8 10
#12	8 10
#13	8 7
#14	8 6
#15	5 1
#16	5 7
#17	5 9
#18	5 8
#19	5 13
#20	5 4
#21	5 3
#22	5 6
#23	5 6
#24	5 4
#25	5 6
#26	5 6
#27	5 6
#28	5 3
#29	5 9
#30	5 7
#31	5 9
#32	5 2
#33	5 10
#34	5 6
#35	5 2
#36	5 5
#37	5 5
#38	5 2
#39	2 0
#40	2 1
#41	2 0
#42	2 2
#43	2 1
#44	2 0
#45	2 1
#46	2 1
#47	2 2
#48	2 3
#49	2 2
#50	2 2
#51	1 0
#52	1 4
#53	1 2
#54	1 1
#55	1 1
#56	1 2
#57	1 1
#58	1 0
#59	1 1
#60	1 1
#61	1 0
#62	1 1
#63	1 2
#64	1 0
#65	1 0
#66	1 0
#67	1 0
#68	1 2
#69	~0 1
#70	~0 0
#71	~0 0
#72	~0 1
#73	~0 0
#74	~0 0
#75	~0 0
#76	~0 1
#77	~0 0
#78	~0 0
#79	~0 0
#80	~0 0
#81	~0 0
#82	~0 0
#83	~0 0
#84	~0 0

Lotto Name:	California Fantasy 5
Date range:	January 1, 2023 to February 9, 2026
Total draws:	1,124 draws
	Theoretical Expected Frequency
	Observed frequency from California Fantasy 5's actual lottery draws

Template	Expected Frequency vs Actual Frequency
#1	79 77
#2	79 66
#3	79 77
#4	70 68
#5	40 36
#6	40 31
#7	40 44
#8	36 30
#9	36 39
#10	36 40
#11	32 30
#12	32 41
#13	32 28
#14	32 40
#15	32 48
#16	32 29
#17	23 24
#18	23 18
#19	23 29
#20	21 29
#21	21 27
#22	21 12
#23	21 21
#24	21 20
#25	21 27
#26	16 15
#27	16 7
#28	16 13
#29	11 13
#30	11 8
#31	11 14
#32	11 13
#33	11 9
#34	11 10
#35	8 9
#36	8 9
#37	8 10
#38	7 3
#39	7 5
#40	7 11
#41	4 3
#42	4 4
#43	4 5
#44	4 3
#45	4 6
#46	4 1
#47	4 3
#48	4 4
#49	4 2
#50	2 5
#51	2 4
#52	2 0
#53	~0 2
#54	~0 1
#55	~0 0
#56	~0 1

The odds of winning stay the same

Nothing on this page changes anyone’s odds, and it is worth being blunt about that. The jackpot odds for the games in this study are fixed by each game’s design: 1 in 292,201,338 for Powerball, 1 in 290,472,336 for Mega Millions, and 1 in 25,989,600 for Lotto America.

Those numbers come from the same combinatorics used everywhere else in this analysis. Studying templates, frequencies, or ten years of draw history moves none of them by a single decimal place. That is the honest limit of what lottery analysis can do.

Why the match is never exact

Expected and observed frequencies agree closely, but they never agree perfectly, and they never will. Randomness is streaky over short stretches. A template expected in roughly 7 out of 1,000 draws might show up four times in a month and then go quiet for a while. That is normal behavior for a random process, not a signal.

The law of large numbers only says the observed proportion approaches the theoretical probability as the number of draws grows. It says nothing about any single draw, and it never owes the data a correction. If you want to see this behavior with your own eyes, I built a visual simulation of lottery randomness that shows convergence happening in real time.

Why dataset consistency matters in lottery statistics

Lottery statistics only hold up when the data behind them is clean. Clean data starts with one rule: never mix draws from different game formats.

A lottery game is not always the same game across its history. Number fields change. Pick sizes change. A 6/36 game and a 6/49 game are mathematically different games with different probability distributions and different frequency ratios. Stack their results into one dataset and you do not get more data. You get a polluted sample that reflects neither game, and every conclusion downstream becomes unreliable.

A real example: NJ Pick 6 history analysis

A reader named Maru (not his real name) ran straight into this while trying to verify the math on his own:

Hi Edvin, I bought your 6/46 calculator and downloaded the “prevalent” combination template. But when I check any suggested composition against past NJ Pick 6 results using the search tool for past winning numbers, it never shows as a jackpot winner, even going all the way back to 1980. I just want to check if your math works based on actual historical data.

Checking the math against real draws is the right instinct. But this test had two problems, and neither one is the math.

The first is the dataset. NJ Pick-6 has run as 6/36, then 6/39, then 6/42, then 6/46, then 6/49, and eventually came back to 6/46.¹ Each format is a different game in combinatorial terms. A 6/36 game has 1,947,792 possible combinations. A 6/49 game has 13,983,816. Pulling results back to 1980 stacks five different games on top of each other, and the Lotterycodex templates built for today’s 6/46 format cannot match that mixture. The mismatch comes from the data, not the math.

The second is the method. A template does not describe individual combinations winning. It describes how often a combinatorial composition appears among winning draws. The correct check is to classify each winning draw by its composition and count how often each composition appears across the whole dataset.

Selecting the right dataset

A dataset for lottery analysis should only include draws from a single game format. For the game as it exists today, the earliest valid starting date is the date the current format began, not the lottery’s original launch date. For Powerball, that is October 7, 2015, when the 5/69 format was introduced. For Mega Millions, it is October 31, 2017, when the game moved to 5/70. Everything before those dates belongs to a mathematically different game and should be left out.

You do not have to start on the exact day the format began, either. Any window inside the current format is valid. My own Powerball table on this page starts on January 1, 2020, because more than 865 draws is already a large enough sample to check agreement with the law of large numbers. The same reasoning applies to the other games on this page. The one boundary you can never cross is the format change itself.

How to verify lottery statistics properly

Confirm three things before running any comparison: the game’s current format, meaning the number field and pick size; the date that format was introduced; and that every draw before that date is excluded. Once those conditions are met, the observed frequencies of the different combinatorial compositions gradually move toward their theoretical values as draws accumulate. That is what the law of large numbers predicts,² and it is what real draw data shows when the dataset is set up correctly.

Expect short samples to look noisy anyway. That is the same short-run streakiness covered earlier on this page. And format changes are not the only trap. A change in draw schedule skews per-week counts. A bonus ball change on its own does not affect the main number matrix, but if the main matrix changed at the same time, the dataset restarts from that date. And a poorly maintained results database distorts everything, no matter how careful the rest of the setup is. The math cannot compensate for data that does not match the game’s structure.

About the draw data

The draw data on this site comes from a third-party results feed covering multiple games and years of history. At this volume, checking every single result by hand is not practical. I run regular spot-checks to keep things consistent, but across a dataset of more than five thousand draws, I cannot completely rule out a small number of errors.

If you want to verify anything independently, go straight to the source: each lottery publishes its official draw history on its own website, and that should always be your reference of record.

Frequently asked questions

Can lottery analysis predict the next draw?

No. Each draw is an independent random event. Analysis of past draws, including everything on this page, tells you how a game behaves over the long run. It tells you nothing about the next drawing, and no method exists that can.

What is a Lotterycodex template?

A template is a specific combinatorial composition: a way of describing a combination by how many of its numbers come from the low-odd, low-even, high-odd, and high-even sets of a game’s number field. Because the combinations in each template can be counted exactly, each template has an exact probability, which is what makes the comparison with actual draws possible. The details are in The Lottery Formula.

Why does your Powerball analysis start in 2020 when the current format began in 2015?

Any window inside the current 5/69 format is a valid dataset. The format began on October 7, 2015, and that date is the earliest boundary, not a required starting point. Starting from January 1, 2020 gives me 865 format-consistent draws, which is a large enough sample to check agreement with the law of large numbers. The only rule a dataset can never break is crossing a format change.

Can I use a lottery’s full draw history for statistical analysis?

Only if the game has never changed its format. Mixing draws from different formats produces a dataset that reflects neither game, and the results will not line up with theoretical expectations. Use draws from a single format only, starting no earlier than the date that format was introduced.

How do I check Lotterycodex templates against actual draw results?

Classify each winning draw by its combinatorial composition and count how often each composition appears across the whole dataset. Compositions are what carry measurable long-run frequencies, and their observed counts move toward their theoretical values as draws accumulate.

About the author

Edvin Hiltner

Edvin Hiltner writes about lottery mathematics—combinatorial probability and statistical modeling of lottery draw outcomes. His Lotterycodex Publishing Policy outlines a math-first approach meant to address common misconceptions and encourage responsible play. He's studied probability theory against historical draw data — Powerball, Mega Millions, EuroMillions, EuroJackpot, UK Lotto, Irish Lotto, Tattslotto — since 2017. He built the Lotterycodex calculators in 2019, a PHP simulator using random_int() in 2020, and released free probability tools in 2025. His SSRN paper, The Insufficiency of Single-Dimensional Combinatorial Analysis in Lottery Number Systems: A Case for Four-Set Partition, came out in 2026. None of it is about predicting winning numbers. It's about what randomness looks like once you box it into a fixed system of rules.

References

Lottery Analysis and Statistics: What 12,954 Actual Draws Show

What lottery statistics can and cannot tell you

How the analysis works

Powerball statistical analysis

Mega Millions statistical analysis

Lotto America statistical analysis

More lottery games

The odds of winning stay the same

Why the match is never exact

Why dataset consistency matters in lottery statistics

A real example: NJ Pick 6 history analysis

Selecting the right dataset

How to verify lottery statistics properly

About the draw data

Frequently asked questions

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What lottery statistics can and cannot tell you

How the analysis works

Powerball statistical analysis

Mega Millions statistical analysis

Lotto America statistical analysis

More lottery games

Cash4Life (Retired)+

Arizona Fantasy 5+

Lucky for Life (Retired)+

Gimme 5+

Tri-State Megabucks+

Arkansas Lotto+

California Fantasy 5+

The odds of winning stay the same

Why the match is never exact

Why dataset consistency matters in lottery statistics

A real example: NJ Pick 6 history analysis

Selecting the right dataset

How to verify lottery statistics properly

About the draw data

Frequently asked questions

Leave a Comment Cancel reply