Pattern Analysis of MegaMillions Lottery Numbers

You’ve collected and summarized a bunch of data here, but it would be much more useful (and straightforward) to run some inferential procedures (Monte Carlo procedures would be particularly easy to implement in this case) to see if these results were compatible with the hypothesis of a fair game. Give it a try!

Hi, first of all congratulations on your deep analysis.

I made something similar on the Spanish Lotto may years ago but using a different approach.

I started asumming the game was fair and the lottery wasn’t rigged. Then I compared every winning combination (and sub-combinations of 3, 4 and 5 winning numbers) with the expected number of winners for the total bets on each game each week.

Using this method I confirmed a theory many people (myself included) has: that not all the players play their numbers "randomly" but that they have some "favourite numbers" and others that people just don’t like. Combos like 1-2-3-4-5 are played a lot, also date combinations — while numbers from 31 seem to be played less — leading to too-many-winners or no-winner scenarios in each case.

You can detect this by checking how many winners are in each prize category on each game, comparing with the expected statistical results (not only for the big prize but for the smaller ones also).

On the Spanish Lotto (a basic 6/49 lotto, with 1 in 14 million odds) sometimes you have no big prize winners even when there are 30 or 40 million bets (and there "should be" at least 1 or 2); sometimes you may have 10 or 20 winners with only 10-15 million bets (for an «easy» combo like 1-2-4-8-9-24 or 7-14-21-28-41-42).

People can select their own numbers or let the «machine» select it at the shop, those numbers are supposed to be random and doesn’t seem to influence on this (even if they are large).

If you read Spanish or can find a good translator you can find my (totally amateur) work here

http://www.microsiervos.com/archivo/azar/loto-un-sistema.html

— Alvy

I have read the article and would like to ask if you think that the results can be used to predict any trends in future outcomes of the lottery?

If not, what test would be necessary to prove that the balls are not evenly distributed?

Have you considered doing a REAL STATISTICAL TEST instead of making graphs. I doubt any of this data is statistically significant.

Assuming a completely fair game could a player still benefit from playing numbers that were played less frequently by other players? For example, you could play more numbers higher than 31 which are not dates, so there are fewer other players to share a prize with, assuming other players play 1 to 31 more frequently. Not only would the winning numbers be looked at but also the number of winners for a given combination or number.

Turns out I did this exact same analysis (but i didnt think to write it up, nor that anyone would be interested in reading it).

At the time I did it (well over a year ago), 32 was the most common winning number and 3 was the most common mega ball.

The problem than, and the problem now, is that there havent been enough drawings to see that the histogram is truly flat or not (truly random).

I would suggest that, the drawing is in fact random, and choosing least common numbers will be better than choosing more common numbers. the reason for this is that eventually we would expect the histogram to be flat, in time the occurrence of infrequently drawn numbers should increase so as to flatten out the histogram.

However, each drawing is random. Before we see another 35 (the least common number when I did it), we could see 20 more 32s.

I read the article and the first thing I noticed is that he didn’t account for the number of times that there wasn’t any mega millions winner…you know, how many weeks went by without a jackpot winner…where are those "no winner" numbers in his analysis?

Has anybody seen this – it’s about using mixing errors in lotto picks:

http://use4.com/Prove-it.html

In what area did you include the "luck" factor? The statistical analysis appears complete; except… in all areas of gathered factual information there tends to be a human factor skewing the results in one fashion or another. Might the human factor in your analysis be "luck"? After all, it appears that some people are simply luckier than others.

Does anyone have data showing the distribution of tickets vs winners? I remember, a while back, that it was skewed towards Georgia for some reason…

Some lottery corporations are changing the balls set over time.

This is great work, thanks for posting this. I’ve done some similar ongoing analysis, and created a website http://www.lottoroller.com/megamillions for both MegaMillions and PowerBall.

I can claim myself as a Powerball winner and it was all luck. My winnings amounted to 3500 dollars and it was a bonus drawing ticket and had nothing to do with picking the right numbers. That being said, I lived in Las Vegas for seven years and played a lot of video keno which is just like the lottery. I won some jackpots but lost more than I won. I won on single screen keno and I won on multiple screen keno which is like playing up to twenty lottery number combinations on one game. I probably played well over 100,000 games of keno or more and I tried every trick in the book. What I found is that there are two ways to win without a lot of luck and that is 1.) to overwhelm the game with number combinations. Or 2.) Do like the 80 year old man who recently won powerball and play the same numbers every week for 17 years (which is just what he did). The first option is expensive and the second option relies on the possibility (not probability) that if you stay on the numbers long enough, they will hit (as is the case in Keno). Take your pick.

I recently performed some simple statistical testing on the data set corresponding to the current rules with 56 draw and 46 bonus balls. The data appears to exhibit acceptable variance, randomness and normality based on Chi Squared, runs test for randomness, and Anderson-Darling test for normality. The upshot, is that no matter how you pick your numbers, you have 1 in 175 million chance of winning. Best of luck.

Interesting line of thought regarding your data, matthew. I’ve also been down this road, but have since abandoned such trend/frequency analyses. However, i found a theory online specifically addressing the true randomness of lotteries when a mechanical means is used to generate the results, e.g. balls bouncing about in a glass box via tumblers, air, etc. What this theory, which i call “the mechanical selection of randomization” (since i don’t remember what the author called it) did was confirm i was seeing patterns in something allegedly random. simply stated, the theory hypothesizes that if a mechanical process is used to select something randomly, and there are no variables introduced to alter the natural mechanical process, unintended PATTERNS inherently occur over time due to the repetitive nature of the mechanical process. i believe the theory is sound and lotto commissioners took note of it, also. if you’re on this thread, i assume it’s because you’re also fascinated by the alleged randomness of a lottery, or you see peculiarities in lottery results just by reading the actual winning numbers in chronological order. nevertheless, as i’ve experienced through the type of research mr. vea shares, seeing a pattern and predicting one in a 6/42-based lottery is an entirely different matter. for those live lotto draws, wise commissioners could easily thwart folks like us who love to crunch and analyze data simply by changing the order balls are dropped, rearranging the ball setup, increasing the air pressure for applicable systems, letting the balls tumble about longer, etc. “weighted balls”, imo, raises the specter of impropriety and corruption. then there are those backroom lotto draws that mys tate either does or used to. for example…if folks like us knew every lotto combination purchased for a particular play date, we could then determine which combinations weren’t purchased. if the draw isn’t live, a lottery commissioner, with the aforementioned knowledge could conceivably control WHEN and WHAT number combinations won…assuming all possible winning combinations weren’t actually purchased. anyway, if someone can predict the next lottery number “pattern” by examining trends and frequencies, that person is a g’damn genius! (or one lucky sob.)

“unintended patterns due to the repetitive nature of a mechanical process undermines true randomness”. it is this thought that motivated me to no longer consider looking at how often a number appears nor even when, but to focus more on GROUPS. it is my theory, piggy-backing on that mechanical selection of randomness theory, that if true randomness is undermined by a mechanical selection process, over time those patterns that develop can be identified through GROUPS. hint: in my theory you must think beyond how many balls are actually drawn to comprise a winning combination. forgive me folks, but i love opining about lotto!

Repeated numbers (( Only one set of five numbers has ever repeated twice in the history of MegaMillions: (11, 14, 18, 33, 48)).Please tell me im wrong But i dont think that any numbers have repeated themself like that AT LEAST NOT AS OF THIS COMMENT. PLEASE INFORM ME IF IM WRONG. lottery pirate

Too bad the lottery does not publish the numbers in the order they were drawn. That would make a measure of central tendencies more acurate to analyze. New York has a FOIL (Freedom of Information Law) which allows someone to request lottery records at $0.25 per page, but then you would have to enter the data in by hand.

Sucks huh?

Maybe the reason they don’t publish it is so it’s harder to make forecasts.

have you studied roulette numbers into lottery it comes alot to win 4 numbers 5 numbers in grandseries and tiers if you look at the numbers are standed tier is 33 16 24 5 10 23 8 30 11 36 13 27

Great work! So with all this info, how much have you actually won? Must be a lot, because only someone with A LOT of time on their hands (like a lottery winner) would come up with something like this.

Great work here on the data analysis. One thought to bear in mind, all the lotto balls are not dropped, thus you may not have a full frequency analysis. This may sound a little strange, but to really get the answers you are looking for you need a test. No self respecting engineer would do this on data alone, you need the physical test to accelerate and refine the data. I saw an exhibit recently that comes to mind, it was at a children’s science fair. It was a giant plinko board about 10 feet tall by 15 feet wide. It released balls at the center point at the top from a tumbler. It was designed to explain statistics and distribution theory. It worked wonderfully as you would get a nice distribution curve, almost perfect, visually impressive. My thought would be to build such a device and drop the lotto balls “numbered” and run the test 10,000 times. Log the data and then give me your comparative results. Two tests would even be better. try dropping all 50 balls and test the repeatable distribution of number sequences. The problem with most folks is we’re lazy and won’t take the time to engage such a project. Plywood, nails, ping-pong balls….away ya go…. Take your own data…..

Sincerely,
Test Engineer

Hey I have to say this is some great data, but I am not sure I understand what we are concluding here. The fact that the lottery is random, does this data indicate that over time the curve becomes normal for particular numbers, or is this saying just the opposite?

The biggest problem you face here is that the sample set versus the possible combinations is very small. To spot an honest trend here would require millions of drawings versus the thousands that have occurred. At least your data doesn’t point to any glaring non-random trends that would cause one to be suspicious. For those who have conspiracy theories about the game just remember there is no centralized data processing of the numbers sold – each state holds their own sold number data base and they all scan after the drawing to find winners and then report that upstream to their collective central control group.

For those of you looking for some edge in this game, there is no mathematical one as the data shows (half the proof being the author published the data!) but there is a way to play this game from a gamblers perspective that is superior to the usual approach.

An example: your gang from work buy 100 tickets each week, everyone throws in 5 bucks or whatever. You could get a “bullseye” win but when your buddy buys the tickets he is going to get 100 quick picks and thats it – in 100 tickets there is a fairly good chance (not high but real) that you dont even get the megaball (or powerball) once.

The lottery slips let you choose a powerball or megaball and have the rest chosen at random. The smart thing to do is this: instead of buying 100 tickets buy just 2×46 tickets so you have 92 tickets (or do 3x at 138). You now have 2 or 3 of every megaball. This guarantees a minimum of 2 or 3 ticket wins AND it reduces the chance of getting 5 numbers WITHOUT getting the powerball to some degree.

Looked at from another angle if you buy 46 tickets at random, they all have 3 matching/drawn/winng regular numbers for some insane reason BUT none of them have the megaball you win like 10 bucks for each. Buying every megaball for sure however turns one of the tickets into a $150 winner guaranteed.

If you intend to buy large amounts of tickets its insane to not buy every megaball because its easy – there are two tanks, one you need to match 4 of 56 numbers from – thats hard, the other you just need to match 1 of 46 – thats so easy you can just buy them all and reduce the game odds slightly by avoiding the “no megaball” outcome.

Your analysis is one of the more thoughtful I have seen online.

You are on the right track using SQL. That could cutdown on programming a bespoke algorithm, but may lack flexibilty in the long run.

I noticed you analysed data from a lottery site. Such data is skewed since no lottery draws are ever sequential. One of the first rules of statistics is boring old observation. Observing each draw is tedious but necessary.

One of the commentors mentions air pressure, ball drop order and mixing period. Mixing periods do vary per draw, but not significantly (in the lottery I study). This commentor has discovered the non-statistical aspect of lotteries which appears to be the inescapable power of geometry and physics, not to mention gravity and trajectories of bouncing balls.

Any real attempt at scientific prediction of lottery results would have to consider all non-statistical elements, as well as statistics based on well-observed data.

The fascinating aspect of lottery predictions are the extensive scientific disciplines that can be deployed to aid accuracy. Some of these are not apparent in prototype system development. Its likely a prediction system prototype for lotteries would take years to develop… even decades.

So, I find myself asking this question:

If I had to search through millions of combinations to find one that had a financial value way beyond what I could earn in a lifetime, would I still continue this search?

Of course most of us would. In a sense, we are biologically programmed to search for value. So the commentor who mentioned more valuable groups (of combinations) is certainly heading in the right direction. Better to search in rich fertile lands, than barren wastes, right?

As in depth as your analysis is, without being able to recreate the drawing procedure exactly to minimize variables, I don’t believe statistical analysis of past results can prove rigged theories or help predictions.

I compared 2010 results and there are certainly odd frequencies that would leave one to think it’s absolutely rigged, but without being able to perform the procedure myself, I’ll just have to consume the data, pray the game isn’t rigged, and pick my numbers.

“The rigging works like this: super computers keep track of each combination sold, and then the ping-pong balls are weighted to assure that a losing combination comes up. On rare occasions, all possible combinations are sold, and they must let someone win. Only then is the game honest” as per the link supplied in the footnotes http://www.uncoveror.com/lottery.htm

This would be more believable if ticket sales didn’t close a mere 20 minutes prior to the drawing. 20 minutes isn’t much time to analyze non-purchased combinations and then weight the balls with a gas.
Aside from that, should the scandal be leaked, (as the recent Manning case) it would be political suicide and the lottery industry would collapse.

Terrific article and great work. I think the information here will lessen the odds of losing as you (we) know what not to do. Now I have to get back to work and study the previous years.

By the way, how long did it take you to gather this information and form your stats?

Good luck,
Joe

Using Excel for statistics is a big no-no. A proper statistical package, like R or SPSS should be used for this sort of things.

Also I don’t even see a proper statistical test here, I don’t know a chi square test against an uniform distribution of values would be a good start. Looking at the SD deviation from the linear regression of the counts is not very informative.

Does anyone know if Mega and/or Power Ball use more than one set of balls for the drawings. When I lived in AZ years ago, their state lotto had 3 or 4 (can’t remember exact number) different sets of balls. The sets were selected randomly before each draw.

How about the 19 y.o. who had a dream about 6 numbers and never played the lottery in his life? He had no idea why he kept hearing these numbers and his Mother told him to play the lottery with these. He won $35million. Or the two women(separate winners) who visualized winning or praying about it? Or the people who use birthdates/anniversaries? Some say it’s the time of day that you buy your ticket.

It would be nice to know the answer and even better to win!!

Drawing balls are retired after a specific number of months, or earlier if the balls display any type of wear. Off camera, in front of an auditor two random envelope drawings are completed to choose both the drawing machine(s) and ball set(s). If there are any sort of patterns appearing in the envelope draw the auditor can ask for a redraw. Typically 2-3 on camera test drawings occur before each official draw (all in front of the same auditor). This is done to verify that the random machines are working correctly and that random balls are producing random results; if they are not the auditor can request additional testing, inspection, or even another random envelope selection. Machines are cleaned daily in front of auditor after official draw. Quarterly a physical inspection of each ball is performed, if one ball in the case is damaged the entire case is replaced with new balls. All those things to consider, the important factor is that balls get replaced – long term analysis on ball draws may be less accurate/effective than short term. This assumes that you know when a ball set has been replaced.
Oh, one last thought for anyone who thinks it might be rigged no one gets near locked machine/ball storage room alone without auditors; and if there is a fire alarm before drawing you have to protect the machines/balls first and bring them with you outside in front of auditor.

I’ve played PowerBall for years now and use a very simple approach. I purchase a $5.00 ticket and select all 2’s. Odds are in my favor that the 2’s will eventually be drawn.