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So there are a few people mentioning that the chances of this happening are equal to the chance of any other set of 6 numbers coming up, but something I like to think about is how to estimate the size of the set of winning combinations that would trigger this kind of reaction, and what the chances of one member of that set being the winning numbers would be.

So let say, a set of:

- sequential numbers.

- sequential primes.

- sequential even or odd.

- a commonly memorized multiplication table 3,6,9,12...

- squares 2,4,8,16,32...

- other famous sequences - eg fibonacci

- famous numbers - eg 4,8,15,16,23,42

- the same numbers being picked multiple days in a row

As a bonus, I like to think of the number of lotteries important enough to warrant making the news. Then you can calculate how frequently you can expect to see a 'crazy lottery winning combination' story.

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If you take OEIS as a database of “interesting sequences” (some of them are only interesting to mathematicians), and pull out all of the running groups of 6 (in the sequence as provided, which of course doesn’t include the full version of very long or infinite sequences) that could be a powerball number, you get 1,097,698; or about a 2.6% chance; just the first 6 digits of each sequence gives a mere 32,110; or about 0.076%. Of course most of these aren’t actually very interesting, after all; but it does provide a rough approximation.

(Code here: <https://gist.github.com/wolfgang42/2df001b05065488620700f0fd...>)

I think this is an important point. I think someone asked "what's the probability this number is chosen?" and many people respond with 1 in 42 million or something like that. But in fact this is only true under the assumption the lottery is implemented perfectly. But this assumption is what is under scrutiny here. Because the lottery is not some ideal process in the mathematical universe, it's a process in the physical universe implemented by humans. Because it's a real process implemented by humans, there's all sorts of ways we can imagine the results could be skewed by the implementation details, and we can imagine all sorts of outputs that might be more likely given the most likely flaws in the implementation.

Yeah, it's not a question of what is P(X=5,6,7,8,9), it's P(Someone is cheating | X = 5,6,7,8,9).

In any case, there's probably a few people out there who picked "1 2 3 4 5" that are kicking themselves extra hard.

True, but it’s random enough to make analysis difficult.

I’d be more worried about lack of transparency and poor controls on the part of the game administration. The secretive organization that runs one of the big lotteries requires NDAs for everything and is super secretive, but lacked internal controls to prevent an insider from rigging the game.

I would think it more likely people pick days of the month and month numbers (birthday, wedding day, etc). That hypothesis can be tested by looking at the number of winners vs the winning numbers. Draws where all numbers are over 31 should have relatively fewer winners.

Also, if the numbers have to be picked from a grid, the layout of the form may drive what people pick (similar to why 2580 is a relatively popular security pin. See https://www.datagenetics.com/blog/september32012/)

And interesting squares you picked there :-)

I used to work at a lottery company and I can confirm your hypothesis. Numbers below 31 and magical numbers like 7 or 13 are picket more frequently. In a game where we draw 5 numbers out of 90 the combination 1,2,3,4,5 was played 4-5000 times a week on average, which means 1 lottery ticket in every 3000. Which is a bad strategy in games where the pot is splitted up equally between the winners. You could have statistically duoble your expected win (return on investment) picking unusual numbers. But ROI was still at 70%.

[Edit] We had a lot of funny stories dealing with obsessed players. One of them accused us of cheating because he found the winning numbers in the newspapers of the last week, in the section of financial news. He also sent us copies of the newspapers, filled with encirceld numbers. It remembered me of one scene in the movie A beautiful mind where John Nash does the same with words.

> funny stories dealing with obsessed players

aka sad stories of the mentally ill people your company was exploiting.

Lottery: A tax on people who are bad at math.

- Ambrose Bierce, The Unabridged Devil's Dictionary

I've done the math on my local lotteries and the EV tips into the players favour when the jackpot gets a bit bigger than 30 million, even if you share a jackpot with one other. This is due to the return of 3,4,5,6 and 7 numbers all adding up, combined with the fact that the jackpot is largely from what others have spent in previous weeks.

If I was completely rational I'd bet my house on it when it tips into my favour. I am not completely rational.

>If I was completely rational I'd bet my house on it when it tips into my favor. I am not completely rational.

It would be rational to bet your house on it if you could repeat the draw an infinity of times, so that the outcome would converge to the expected value.

A rational actor would consider risk of ruin.

Not wanting to protect the company, but it's called the National Gambling Company, so it's pretty obvious for everyone what its products are, the participation in the games is anonymous, the customers have the opportunity to opt out (not buying lottery tickets) and it is owned by the government therefore it's profit is nothing more than a special form of tax. Their games are supervised directly by the Ministry of Finance.

I could came up easily with a bunch of well know companies/industries breaking all these rules at once, but that's an other topic.

Even if these "noticeable" sequences make up only a tiny fraction of all possible sequences, the probability that they show up at least once in a while among all of the world lotteries is actually high.

In general the probability that some unlikely events occur is high.

I think it's still not that frequent. More frequent than I expected before running the numbers, though. Let's say you have a five number interesting sequence starting at N. If the highest number that you can draw is M, then at most, you will have M-5 of these possible. Call it M of them. But we probably don't have sequences starting at each N.

Let's say we've got k of these different kinds of interestingnesses, and an average sequence can start at like a quarter of the numbers. Then the number of draws that we would consider interesting are no more than 0.25 * k * M. So the probability of an interesting draw is 0.25 * k * M / (M choose 5). If M = 69 (apparently the PowerBall rules), then it's 0.25 * k * 69 / 11238513 = k / 651508.

The probability of drawing one of these in c draws is 1 - (1 - k / 651508)^c. For a draw to be at least 50-50 (where it becomes more surprising to not have seen one, that's at least -log(2)/log(1 - k/651508) draws.

For 5 interesting sequences, that's about 90k draws. For 10 interesting sequences, that's about 45k draws. For 100 interesting sequences, it's about 4.5k draws. By 100 sequences, I think the number of numbers the sequence is eligible to start at will drop by a ton. Even having a quarter of numbers be eligible start positions with 10 really interesting sequences seems like a tall order.

So I'd guess the number of draws for a "real" answer is between 5k and 100k draws.

Looks like about 50 U.S. states and territories do it 2x / week. I don't know internationally, but let's double that number to 100 places, 2x per week: 200 draws per week.

5k/200-100k/200 = 25 - 500 weeks = 6mo to a decade before this isn't surprising. I'm leaning more toward the decade end.

Just realized it’s six balls drawn not five, so it’s one in about 7M per draw, nearly 10x less likely, so it would take about 10x longer, somewhere between 5 and 90 years before it’s unsurprising. So if the number of draws per week is accurate, this is reasonably suspicious.

Thanks, I was about to write about exactly this.

The list of squares looks more like powers of two btw ;)

Ha, you're right. Leaving the mistake in. Appreciate the correction. I was going to do perfect squares but switched to powers and didn't relabel. :facepalm:

This kind of analysis reminds me of the classic SSC post, "The Pyramid And The Garden" https://slatestarcodex.com/2016/11/05/the-pyramid-and-the-ga...

Depending on how you count, famous numbers isn't exactly a well defined category, the number of special sequences will be less than 100 and certainly less than 1000. Regardless, very small compared to the overall number of sequences.

I’m not great at maths, but back of fag packet let’s say there are 100 suspicious sequences, and there is a 1 in 15 million chance of winning the lottery - that’s 100/15000000 or 1/150000.

Let’s assume that there are on average 2 lottery drawings per week per country, which means c400 lotteries.

So 400/150000 = 1/375 chance per week, so it’s going to happen every 10 or so years.

Your math is correct. I would say two draws per week is wrong but I'm not a lottery expert. Every 10 years may seem surprisingly often, but we are including sequences like 2,3,5,7,11,13 which most people wouldn't find special. Even 10,12,14,16,18,20 (double of this "special" sequence) I think a lot of people wouldn't care as much.

Do you think two draws a week is a little on the high side or the low side?

The U.K. has 7 draws per week, and I assume the USA has draws in every state, so it could add up fast. I’m assuming most lotteries are weekly, but they could be much more infrequent.

I assume there are other funky possibilities like the crazy thought that someone could get their own phone number as a result, or the sequence 10,20,30,40,50,60 etc so I suspect there are a few other notable sequences hidden in there.

In Poland we have a lottery that draws 20 numbers out of 80 TWICE A DAY. Other lottery is drawing 3 times a week. So a lot of drawings.

EDIT: Oh, the best part: we have this "lottery for the impatient" that has draws every 4 minutes :)

It's called Keno, it predates most government-run lotteries, and many casinos run it every few minutes. The govt lotteries simply co-opted the game.

Can we define an order of "specialness" on the set of sequences? Because in this case, you have a subset of the non-special sequences which are the least special of all. Which makes them special.

The “interesting number paradox.”[0] Basically, if one were to classify numbers into two categories: “interesting” and “uninteresting”, you’d end up with a contradiction as the smallest “uninteresting” number is itself “interesting” for being the smallest in the set.

[0]: https://en.wikipedia.org/wiki/Interesting_number_paradox

Ah thank you I was looking for the name and couldn't remember it.

Preventing self-reference until instantiation resolves this paradox. Am I missing something?

No, you didn't miss anything, apart maybe the fact that I wrote this tongue-in-cheek. ;-)

Kolmogorov complexity is one metric: what is the size of the smallest program that can produce the sequence?

(Note that Kolmogorov complexity is not generally computable, because you could solve the halting problem if you can compute it for all sequences).

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