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South Africa's lottery probed as 5, 6, 7, 8, 9 and 10 drawn (bbc.co.uk)
220 points by EwanToo 3 months ago | hide | past | favorite | 408 comments



Does anyone have any idea how many daily "official" lotteries are drawn worldwide?

I'd guess something on the order of 1,000 maybe? (One per country, plus a bunch more for individual states within countries?)

I know they vary in numbers of positions and values, but I am kind of curious roughly how often you can expect a sequence of consecutive numbers (increasing or decreasing) to be chosen anywhere worldwide.

As an initial guess, if it's 5 balls drawn from 50, a thousand times a day, then it's something like a ~13% chance to happen somewhere yearly.

So since this is international news, and not the kind of thing that gets reported every year... it doesn't seem implausibly unlikely, no? (It's not like it's a once-in-a-millenia or once-in-earth's-lifetime kind of thing.)


They draw 6 balls out of 50. There are C(50,6) = 15,890,700 possible draws (C(n,k) = binomial coefficient)

45 of the draws are made of 6 consecutive numbers (1-2-3-4-5-6, then 2-3-4-5-6-7, etc, until 45-46-47-48-49-50)

A single draw has a 45 out of 15,890,700 chances of being 6 consecutive numbers

A single draw has a probability of 1-(45/15,890,700) of NOT being 6 consecutive numbers

Assuming 1000 draws (lotteries) per day, in a year we expect probability (1-(45/15,890,700))^(1000*365) = 36% that none of the lotteries draw 6 consecutive numbers

So there is a 64% probability that at least one lottery will draw 6 consecutive numbers in a year. If there are 1000 draws per week (instead of per day) the probability is still 17% that this will happen in a year.

So this South African draw is kinda expected.


According to a comment elsewhere in this HN discussion, it seems there are rather something like <200 draws a week. That gets us to just a 3% chance of this happening in a given year. That's still quite a bit more than one might intuit, but not quite as drastic.

Of course, as others here already pointed out that there are far more "suspicious" patterns (e.g. other arithmetic progressions like 5-10-15-20-25-30, primes, ...). And of course they often play by different rules, etc.


3% chance in a given year. And for how many years when it didn't happen have we been running this many lotteries? Probably right around 33.


Not to mention the reverse consecutive numbers, the same numbers, the +=2 numbers, the -=2 numbers, the prime numbers, and all the other sequences that people would find "suspicious" because "omgfwow that could never happen"


These six numbers could've been drawn in any order.


5-10-15-20-25-30, 1-2-3-11-22-33; yeah there are a bunch more that people would find suspicious.


Wikipedia says that powerball is out of 69 balls, not out of 50, and in the U.S. there are only about 100 draws per week. I cant imagine the worldwide numbers are more than an order of magnitude higher (I guessed it at 200/week in a sibling comment). That’s 7.5x more potential draws, and 5x-35x fewer draws per week, which would change the numbers quite a bit, making the weekly chance way way lower than your estimate.


I did check this South African powerball. They have 50 balls.

If they are 200 draws per week, the probability is 26% over 10 years.


You compete (as understood from the article) in Powerball by 50 choose 5, 20 choose 1. So 1 in (50!/(5!⨉45!)⨉20 = 42.375.200. There are 15 ascending "suspicious" winning sequences, as the choice of Powerball is always last and thus must be 6-20. Therefore the chance of winning Powerball with a "suspicious" sequence is 15 in 42.375.200. If there are 1000 Powerballs drawings per day for a year, that gives a chance of approximately 12% of happening at least once a year.

https://www.wolframalpha.com/input/?i=binomial+calculator&as...


Without going into detail about why your math is flawed, which it is -- a basic viability check would be to consider: if there was a 64% probability of some event happening each year, don't you think in your lifetime you'd have seen quite a few instances of that thing happening?


If you read a daily digest of global lottery results, yes!

This might not be in the news because it's so rare. It might just be a slow news day.


...or "crazygringo's" 1000 per day guess could be way, way off, and anybody, like "mrb," accepting it blindly as a basis for their own calculations is guilty (of failing to think critically) by association


There is no 64%. It's hypothetical if there were 1000 draws per day. If you want to proof their math wrong please do more than calling it flawed.


He states a conclusion -- "So this South African draw is kinda expected" -- referring to THIS DRAW, not a hypothetical example.


>probability of some event happening each year, don't you think in your lifetime you'd have seen..

No. This is not particularly newsworthy. But for most HN users and this event the answer is: no, but I did see it on HN and it will probably get re posted cyclicaly ;)


If you have the time,would you mind expanding on this with the detail that you mentioned?


Why is the maths wrong?


Maybe he is referring to the fact that not all of the lotteries have the same structure? Idk it still seems like a good estimate to me


64% likelihood seems like a good estimate to you? Just take a second and consider whether you'd take the following bet: Your $1000 against my $5000 that anywhere in the world in 2021, a state-run lottery (with at least 5 balls out of at least 50, and a sixth out of at least 20) will have a perfectly sequential result.

You're being offered 5 to 1 odds on a probability that you just estimated to be significantly better than a coin flip. Are you in? Or does your gut tell you that 6 perfectly sequential numbers is crazy unlikely?


It’s crazy unlikely for any particular lottery but there are a ton of draws and a ton of lotteries, and it only needs to happen once in a year. I probably would take that bet if it were feasible but then again I like to gamble.


Terrific, the same comment I replied to konjin applies to you as well. I'll absolutely make this bet if there's a trustable platform.

Just curious, if a random HN commenter said, there's about a 64% chance that at least one human being will grow to 12 feet tall in 2021, would you take that wager also? I mean, there are a ton of people on the planet, and it only needs to happen once.


The difference is it’s biophysically infeasible for any human to mature and live as a 12 foot tall adult, considering there is a very small number of 7 feet tall people in 2020. Whereas there is no such constraint for any given sequence of numbers if the lottery allows for them to be drawn. I would rather wager on whether you will continue being an annoying pedant. I prefer my odds there.


Sure, how do you want to do that bet?


I'm 100% serious about making this wager, I've sought out the wisdom of HN for help with a "smart contract" to guarantee trust.

https://news.ycombinator.com/item?id=25306489

At the same time, I'm not a thief -- to be entirely certain that you want to make this bet, have you read the entire thread, where alternate probabilities (far less than 64%) are proposed? You're aware that the original assumption of 1000 lottery draws per day was a guess from thin air with no research nor basis in fact? You're aware that mrb's calculations overstated the number of possible conforming sequences and undercounted the total number of potential outcomes? If you acknowledge that you're aware of all of this and still want to proceed, let's do it, no joke, I'm 100% in.


It starts with the literal first sentence of the comment -- "They draw 6 balls out of 50." No, they don't.


thats... not math.


calculating C(50, 6) is, as is everything that follows.


There are 6 numbers, so it's not a 5 out of 50 lottery. 5 out of 50 means every 42000th tickets wins, which is a very high number, and can only be used in a lottery with small prizes. Of course, there are many lotteries for small prizes, for example there are millions of rounds of roulette plays occurring daily, and of course it's pretty often to have 2,3,4,5 and 6 occurring in sequence on some roulette table somewhere during a day. The phrase "1000 a year" relates to large national lotteries, where probabilities are much much smaller, it's about 1/250mln for US megamillions lottery, for example. So for 1000 of 1/10mln lotteries this will happen once every 273 years. I would say the probability of a foul play is high enough to warrant an investigation.


If someone was going to rig the outcome, why would they make it such a conspicuous number?


You're going to share the jackpot with dozens of other winners, so it wouldn't be immediately obvious which of the winners rigged it. While if you pick a random number sequence and are the only winner, you'll be the prime suspect if anyone realises the draw was rigged.


As a prank and/or to expose how unsafe the system is.


Good point, I hadn't considered that. Seems pretty plausible.


-I'd say that is plausible if next week's draw turns out to be 1-2-3-4-5-6, for instance.

(That is - as is pointed out upthread, such 'weird' draws will occur naturally on occasion. It would, however, be quite suspicious if it happened twice in a row.)

I am sure one of the things the lottery looks into is how many coupons with 'weird' numbers chosen like this week's winning numbers have been picked in the past - of course, if you get a result which makes you suspect shenanigans AND the number of coupons with such a strange sequence of numbers is way up from normal, then perhaps that suspicion is well founded.


true ... might be an innocent mistake too.


mistake? As far as I can see, it could only be either intentional or the result of randomness.


It could also be an attempt to rig gone bad


Sure. I’d call that the result of an intentional action, but maybe that’s just personal preference.


There might be some error in the randomization process that caused consecutive balls to be drawn.


That’s a fair point. In that case it would be the result of a both a mistake and randomization.


software bugs are neither intentional not random


"Does anyone have any idea how many daily "official" lotteries are drawn worldwide?"

https://www.gamblingsites.org/blog/how-many-different-lotter...

Looks like a reasonable analysis. Their total is 180.


Are those lotteries with daily drawings though? They seem to list lotteries, but not specifically those with daily drawings.


At least those I know in Europe and US are weekly...


Assuming for simplicity that it's 6d50; you'd get 6 numbers ascending or decending:

  You have: 50^6 / (50*2) / (1000/day)
  You want: yr
    * 427.79832
about once every 4 centuries.


It turns out 50^6 is much, much bigger than C(50,5)*20 (as the article implies), by about 400x.


there are 6 balls drawn in the example, which would drop that number down to .26%, which would be once every 400 years. I didn't check the rest of your math though.


Cool, as a rough approximation, it is close to (1/50)^6, right? But couldn't the first ball be any ball? That doesn't really matter. So wouldn't it just be 1/50^5 for 6 balls?

And wouldn't the odds really be: p = (1/49 * 1/48 * 1/47 * 1/46 * 1/45) for the balls to be drawn in order in one lottery?

The odds for them not to be drawn in order are then: 1 - p.

If there are 1000 lotteries per day, the odds it doesn't happen in a year are: ynp = (1-p)^(365 * 1000).

The yearly probability would be: 1-ynp.

If the balls can be drawn in any order, but they have to end as 6 ascending balls - then I think the probability is much higher, right?


> Cool, as a rough approximation, it is close to (1/50)^6, right?

No. 50^6 is about 938x bigger than (50 choose 6), which is the correct chance of picking 6 of 50 numbers.

> And wouldn't the odds really be: p = (1/49 * 1/48 * 1/47 * 1/46 * 1/45) for the balls to be drawn in order in one lottery?

This is not correct either. There are 45 sequential possibilities of winning numbers: 1-2-3-4-5-6, 2-3-4-5-6-7, ... 45-46-47-48-49-50. So "45 / (50 choose 6)", about 647 times more likely than that calculation for p.

Of course, if you start adding in other striking patterns like 2-4-6-8-10-12, the chance of a "fishy" draw becomes more and more likely.


> Cool, as a rough approximation, it is close to (1/50)^6, right? But couldn't the first ball be any ball? That doesn't really matter. So wouldn't it just be 1/50^5 for 6 balls?

It's very, very far, between non-replacement (~1.36x), order (5!=120x), the fact that final ball is out of 20 instead of out of 50 (2.5x), and the number of possible sequences which would be notable (at least 16x, but actually probably a fair bit more because recognizable patterns are pretty broad).

The upshot of which is that estimating at 50^6 is off by multiple orders of magnitude...


If it's 5 specific numbers drawn from a pool of 50 in any order, we're talking 2118760 combinations of numbers that can be drawn, i.e. a 0.0000472% chance.

For 5 drawn in a specific order from a pool of 50, we're talking about 254251200 ways 5 numbers can be drawn, with a 0.0000004% chance.

In this case it was actually 6 numbers, which is about 10x less likely to match a combination (orderless) and 50x less likely to match a permutation (with order).


The chance for six ascending balls in a single drawing.

First definition. Let's count a sequence as ascending if the sorted set of drawn balls is ascending. E.g. 18,15,14,19,16 with a powerball of 17 would be counted as ascending.

I found that it was easiest to start with the odd one out and work from there. The powerball can be chosen in 20 ways. If the powerball is 1, then the sequence must start at 1 (1 combination). If it is 2, then the sequence can start with either 1 or 2 (2 combinations), 3 means 3 combinations, 4 means 4 combinations. For 5 and above, there are 5 combinations (the upper part of the range has no similar issue, since the normal balls go all the way to 50). We get a total of 85 combinations.

Now every combination of the 5 normal balls can be permuted in 5! ways, so we have 85 * 5! = 10200 ascending draws.

The total number of possible draws are 50 * 49 * 48 * 47 * 46 * 20 = 5085024000.

So the probability of an ascending draw is 10200 / 5085024000 ~= 2 in a million.


Random but there was an incident where the lottery number was identical to last weeks', just out of sheer luck.


Does someone mind posting the math on this? I'm getting something like 775% chance it would happen yearly, but that seems too likely, and I think I screwed mine up somewhere, and wouldn't mind seeing where I made some wrong assumption in how to compute the probabilities.


> I'm getting something like 775% chance it would happen yearly, but that seems too likely,

In probability, when you roam outside of the 0-1 (or 0-100%) range, you can guarantee with 775% certainty that something went wrong somewhere. ;)


That’s not necessarily accurate if you are already operating in probability per year in this case. It just means 7.75 occurrences per time.


You can have an expected value E[X] = 7.75

You cannot have a probability P[X ≥ 1] >1.

There is no 'probability per year' (if it were meaningful year would have to be <1 for a >1 result anyway) - that time frame is built into 'X', it's 'inside' the probability.


If they are independent, then there should be a log-probability of it not happening per year though, right? Like, if the chance for each year is the same and they are independent, then the log of the probability that it doesn’t happen in a given year, If you multiply that that by the number of years, that will give the log of the probability that it doesn’t happen in any of the years. This seems like it could serve many of the purposes that people are intuitively looking for when they think of “probability per year”. And if the different years aren’t independent, uh, maybe some terms could be added to adjust for that?


But OP didn't use either of those terms, they used "775% chance per year", which you can interpret colloquially as a speaker of English.

It's been clear throughout this thread what was meant.


> "chance per year", which you can interpret

Probably something like reciprocal half-life; 775% / yr is about 1 / 1.5 month ie, happens about once every month and a half.


That would be the expected value E(X), not the probability P(X > 0). The two are not conflatable.


Nope. Bad maths. Average of 7.75 per year !== >100% of it happening at least once in one year.

There’s still some chance it doesn’t happen in one year.


Probability it happens is 1 in 50c5 (the number of ways to choose 5 balls out of 50.) So probability it doesn't happen is 1-1/50c5. Probability it never happens in a year is that number raised to (3651000). That gives 85%, so there's a 1-.85 = 15% chance that it does happen each year:

https://www.wolframalpha.com/input/?i=%281-1%2F%2850choose5%...


This analysis is spot-on under the assumption of 1000 lotteries per day, although your asterisks are getting trashed by the markdown.

To the other folks ending up with some wild results, there is a basic checksum on probability: If you compute the probability of an event happening at greater than 100% you've borked something up.

Bug: They drew 5 balls from one pool of 50 with an independent draw from another smaller pool of 20, so you need ~~50 ncr 6~~ (50 ncr 5) * 20, not 50 ncr 5.

Nit: I would rephrase your answer that there is a 0.85% chance that it happens one or more times in any given year. There remains a (vanishingly small) chance that it happened on every single random draw during the year.


Wait. Why is it 50c5? If you want to know how many ways you can roll triples with 3 dice, it isn't 6c3, right?

There's 6 possible triple dice combinations and there are 6^3 possible dice combinations, so it's 6/(6^3).

If you wanted to know the odds of rolling 3 dice in order, you could roll: 1, 2, 3 OR 2, 3, 4 OR 3, 4, 5 OR 4, 5, 6 - which is 4/(6^3) - which is not 6c3.

Why is it different with the lottery? Or did I get the dice wrong?

Or are you calculating that the balls can be drawn in any order?


Balls drawn in any order, without replacement.

The probability of guessing all 6 balls in a single lottery is 1 in (50 ncr 6). So, the probability of losing is 1 - 1/(50 ncr 6). The probability of losing every time is (1 - 1/(50 ncr 6)^(n_games), where n_games = 365 * 1000. Therefore, the probability of winning at least one game is (1 - (1 - 1/(50 ncr 6)^(n_games)).


Got it. 50c6 is the total combinations. But there's more than 1 combination of 6 ascending balls, right? Why is it 1 in (50c6) instead of 45 in (50c6)?


Oh, sure. There's a number of different suspect or convenient sequences out there. All of the evenly-spaced sequences could be considered suspicious if you go broadly enough. A detail I tried to add back in up-thread: The sixth ball is from a separate pool of 20 balls. So 50c5 * 20 is the total number of possible draws, and there are 14 directly in-order sequences.

But the main point was the methodology. (1 - (1 - chance_of_sequence)^(n_draws)).


Thanks! Why are there only 14 in-order sequences??

Couldn't there be 1 2 3 4 5 6 AND 2 3 4 5 6 7 AND 3 4 5... Doesn't this give you 45?


As the person above said, one of the balls is restricted to only 1-20. This ball is drawn last.

You get 14 possible draws if the order the balls are drawn in matters (14-20 being the highest, with 20 drawn last), and 20 permutations if it does not (since any straight above 20-25 is not possible). The math is much different for drawing if the order matters though. There also would probably be some funky stuff going on for the higher straights where order doesn’t matter, since for a 20-25 straight the 20 ball must be the special ball. For a 19-24 straight either 19 or 20 must be, etc. Really you’re looking at calculating “Chance that the first five balls can create a straight with a number between 1-20, and then a 1/20 chance that straight actually happens.”


Sure, sorry. I'd used:

  2*(1/49)*(1/48)*(1/47)*(1/46)*365*1000*(45/50) = 12.9%
The initial "2" term is for consecutive numbers going both directions, and the final (45/50) term is to account for the fact that if you start with 4 or less and decreasing, or 46 or greater and increasing, you'll run out of numbers.

Edit: but if the numbers don't have to be drawn in order (e.g. 8-5-9-6-7 is OK), then the odds are much higher still:

  2*(4/49)*(3/48)*(2/47)*(1/46)*365*1000 = 344.5%
(With the initial "2" term accounting for 4 consecutive numbers on either side of the initial pick -- though I'm not sure I've got that entirely right?) Then it would happen three to four times a year. Even with a 6th ball drawn separately out of 20, that's still a 17% chance happening somewhere in the world in a year, given 1,000 daily draws.


> Edit: but if the numbers don't have to be drawn in order (e.g. 8-5-9-6-7 is OK), then the odds are much higher still:

Most lotteries are order-insensitive, and typically present the results in ascending numeric order. The actual draw often happens in some order (e.g. numbered balls being drawn from a hopper), and it'd be even more unusual if the numbers were actually drawn in consecutive order, but drawing 5-2-3-1-4 would typically be presented as 1-2-3-4-5 and would still be remarked upon as unusual.


To calculate the probability of some event with probability P coming true at least once out of N total tries, you do 1 - ((1 - P) ^ N), not P * N as you've done.

Doing this, your final figures should be 12.1% and 96.8%.

For the expected number of times it would happen though, you are right, it would be three or four times per year on average for the second case.


Oh. I assumed consecutive numbers, but not necessarily consecutive on each pull. That is, 1,2,3,4,5 is no different than 2,3,4,5,1 in my calculations, which makes the possible ways got get consecutive numbers at the end quite a bit higher. Still not positive I got it right, but at least I know it shouldn't match your results, as they are for slightly different things. :)

Edit: What I did was take all the possible combinations of 5 balls (5!) by the number of different sets that could be drawn (50-5, based on lowest number), over the total possible draws (50!). I think perhaps what that does it not account for overlap between sets (1-5 and 2-6), inflating the number somewhat, which is why it's a bit more than twice the probability than you got for any possible sequence of 5.


Unless I'm doing something wrong here - just typing some javascript into the console gives me...

  function rn() {
      return parseInt(Math.random() * 100);
  }


  function drawing(){
      let result = [rn(), rn(), rn(), rn(), rn()]; 
      return result.sort();
  };


  function sequential(arr){
      for(let i = 0; i < arr.length-1; i++){
          if(arr[i] + 1 != arr[i+1]){
              return false
          } 
      } 
      return true
  }

  let counter = 0; 

  for (let i = 0; i < 10000000; i ++){
      if (sequential(drawing())){
          counter ++;
      } 
  }; 

  console.log(counter);

  16
Mathematically, it seems like you'd need to draw one of five numbers from the range, then one of four numbers, then one of three... so the likelihood would be 5/100 * 4/100 * 3/100 * 2/100 * 1/100 = 0.000000012. Although those odds don't seem to line up with the javascript I posted.


FYI 0.129 = 1 / 7.75. I think you and person you're responding to are doing the same calculations, but inverted.


Lumping all world lotteries together introduces a lot of handwaving, so let's start with what we know and work forward.

According to the article, this event happened for South Africa Powerball, where 5 numbers are chosen out of 45, and 1 number chosen out of 20: https://en.wikipedia.org/wiki/South_African_National_Lottery...

That same Wikipedia page does some of the math for us: The chance of one combination being chosen is 1/42,375,200. So if we count all possible sequential combinations, N, we'll know that the chance of a single winning combination being sequential is N/42,375,200.

Say the powerball comes out as any number 6 <= M <= 20. There are 6 ways the numbers 1-45 could be picked such that M is part of the sequence. That's 90 ways total. If the powerball is 5, there are only 5 ways, same continuing down to a powerball of 1 where there's only 1 combination of numbers 1-45 where it could be part of the sequence. 90 + 5 + 4 + 3 + 2 + 1 brings us to 105 as our N.

So this single event had a probability of 105/42,375,200, or 1/403,573. This means that for similar lotteries one would expect to see a sequence after about 200,000 picks.

EDIT: If you only count events where the powerball is the high number, as happened this time, N goes down to 15, making the odds 1/2,825,013, so one would expect such a sequence after about 1.4 million picks.


That seems right, 1 * 1/49 * 1/47 * 1/46 * 1/45 * 5! * 1000 * 365 gets about 8 occurrences per year.


There are two very different questions:

1. How often does this happen? This is a question about expected value, and the answer could be anything zero or above.

2. What are the chances that this will happen within a year? This is a question about probability, and the answer must lie between zero and one. There is no such thing as "a 775% chance it would happen yearly".


Those two questions are closely related here by a very simple transformation: if the expected number of occurrences is N over many independent tries, then probability of 0 occurrences is approximately 1-e^(-N), or 99.96% if N=7.75.

Note that for N close to 0, 1-N is also a good approximation to 1-e^(-N).

For large N, it's generally more convenient to talk about the expectation rather than the probability of 0 hits—I'm sure many readers implicitly converted 775% to the expectation in their heads.


> I'm sure many readers implicitly converted 775% to the expectation in their heads.

Most people cannot do this correctly; the most obvious interpretation of a "775% chance" is that it represents a 25% chance of seven occurrences and a 75% chance of eight occurrences, with no other possibilities.

The problem gets even worse when you have expectations less than one. If the expected number of occurrences is 80%, what are the odds of getting any occurrences at all? They're less than 80% as long as it's possible to have more than one occurrence.


you skipped 1/48


Probability is a 0-1 range


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.


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.


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).


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. ;-)


Setting aside idea of playing a lottery, if you insist on playing it at least do it properly. Don't choose any pattern of numbers as you are exposing yourself to having to share the win with other people. Don't choose a date as it is more probable somebody will also chose a valid date than a completely random set of numbers. Don't choose digits from your favorite irrational number, for the same reason.

Ideally, choose numbers that no other person is likely to choose. Not exactly random. Maybe generate a random sequence and check if you can see a pattern, do a search on the Internet, etc.


This all makes sense, except for "Don't choose digits from your favorite irrational number"... You can choose whatever combination of numbers you want, and find it in your favorite irrational number if you search long enough :)


> This all makes sense, except for "Don't choose digits from your favorite irrational number"... You can choose whatever combination of numbers you want, and find it in your favorite irrational number if you search long enough

You're not reading it right. You don't disqualify digits for occurring consecutively in the expansion of an irrational number. You disqualify digits if the method you used to pick them was to think of an irrational number and extract some of the digits. That is a method that other people can also use.


I’m pretty sure he read it right, the smile at the end indicates he was joking.


> the smile at the end indicates he was joking

This is not a universal convention.


It doesn’t have to be.


And in the instant context, it is not suggested at all.


Good thing we have more than instant context then :)


No, we don't? Perhaps you're not familiar with the meaning of "instant"?


Yeah I guess I don’t. Why don’t you explain it to me? You seem to be quite a smart guy who picks up on everything. Please tell me more about how you are right and I am wrong.


> Please tell me more about how you are right and I am wrong.

Sure. So far you've claimed that (1) the smile at the end of rfonseca's comment should be viewed as good-natured and not mocking. This is not true of such expressions in general, but you've also claimed that (2) you have special knowledge, external to the thread, indicating that (1) is true.

You haven't bothered to support either claim, except to the extent that (2) supports (1); (1) is far from certain and (2) seems extremely unlikely, so unlikely that I surmise you didn't realize what you were saying.

"The instant context" is a common expression which uses "instant" in the sense 4a given to the adjective here: https://www.merriam-webster.com/dictionary/instant


The reason is that there isn't that large number of named/popular irrational numbers and you are likely to choose some digits close to the start (say within couple thousand digits). Good chance somebody else will do the same. I know personally two people who do this (chose consecutive digits from an irrational number).


My favorite irrational number is ϕ(1/10). Can you help me find (1; 2; 3; 4; 5; 6) in it in base 10? ;)

All normal numbers have the property you mentioned, and nearly all irrational numbers are normal, but there are some that are not.


What is the phi function that you use here? If that's Euler's totient, then that does not seem to be well-defined for rational numbers...


Also, the set of “typical” numbers that we actually can prove are normal is quite small.


We can prove that almost all irrational numbers are normal, which is a quite large set. It's just when we pick out a particular irrational number that we have difficulty proving normality (unless that irrational number was intentionally constructed to be normal).


0.101001000100001... is irrational.


Not necessarily. It depends on the combination of numbers (and the irrational number).


Indeed not every irrational number has this property, for instance you could create an irrational number using only the digits 1 and 2, but some irrational numbers do have this property.

Until I did some research a few minutes ago, I thought this property was the irrational number being a "normal number"; however, that is not the case. That all said, a Normal number definitely has this property and I don't think having this property implies normality, but I don't know either way.

An interesting fact though is that almost all real numbers are normal, which means pretty much every irrational number has this property, though not every irrational number. However, we still don't know if pi, e or square root 2 are normal.


> I don't think having this property implies normality, but I don't know either way.

Having this property cannot imply normality. Imagine an irrational number z which has this property, and another number z' constructed from z by taking the first 1 digit of z, appending 1 "2", appending the first 2 digits of z, appending 2 "2"s, appending the first 3 digits of z, appending 3 "2"s, and so on.

Using e as an example, our z' would begin 2.7 2 71 22 718 222 7182 2222 71828 22222...

z' is irrational and shares the property that every sequence of digits can be found in its decimal expansion. But it is obviously not normal; over half of its digits are "2".


Interesting, my first inclination is to be skeptical that the property would still hold, since we're "potentially" slicing up a necessary sequence and adding 2s in the middle.

However infinities are weird* and I think you could construct a proof by contradiction making use of the fact that a sequence of N digits is embedded in infinitely many longer sequences most of which that won't have been broken up by the inserted 2s.

* I'm always skeptical when dealing with infinities and probabilities. Human intuition doesn't gel well with either concept.


Shoring up places where I thought my proof was weaker:

- I rely on the assumption that an irrational number with this property exists. (If it didn't, then the property would imply normality.) This is easy to fix; Champernowne's constant has this property and so the assumption is valid.

- I assert without proving that z' is irrational. We can prove this using the definition of a rational number as one whose decimal expansion repeats after some index. Since z is irrational, somewhere in its decimal expansion there is a digit not equal to 2. (Otherwise, every digit of z would be 2, and z would be the rational number 2/9.) Since z' successively repeats larger and larger stretches of z, this suffices to show that, for any index i into z', there is a higher index j > i such that the jth digit of z' is not 2.

- But we also know that a sequence of n "2"s in a row can be found within z' for any positive n. Assume that the jth digit of z' is not 2. Since we know that a sequence of 2j "2"s occurs later within z' -- it can't occur earlier because not enough digits have yet occurred -- any cycle in the digits of z' cannot yet have begun by index j.

- But since there is no maximum index into z' beyond which all digits are not 2, a cycle in the digits of z' cannot have begun at any index into z'. This shows that z' is irrational.


> Since we know that a sequence of 2j "2"s occurs later within z' -- it can't occur earlier because not enough digits have yet occurred -- any cycle in the digits of z' cannot yet have begun by index j.

This is wrong -- the cycle might begin at j and continue into a huge series of 2s.

But we cannot yet have completed one cycle by index j, and this property can be extended -- there is no index into z' at which one cycle could have been completed, and hence the digits of z' never cycle.

A much simpler proof that there is no cycle goes like this:

Suppose there is a cycle of length n. We know that a stretch of 2n consecutive 2s will appear after the beginning of the cycle. This implies that the cycle consists entirely of 2s. We also know that a non-2 digit will appear after the beginning of the cycle. This is a contradiction; there cannot be a cycle of any length.


> my first inclination is to be skeptical that the property would still hold, since we're "potentially" slicing up a necessary sequence and adding 2s in the middle.

No. If the sequence you want occurs between places a and b of z, then it is a substring of the full sequence between places 1 and b of z, and all such sequences are included within the expansion of z'. (Going by example again, if you're interested in the sequence that occurs between decimal places 41,028 and 315,001 of z, then that sequence will occur within the part of z' that repeats places 1 through 315,001 of z.)


> However, we still don't know if pi, e or square root 2 are normal.

The math is somewhat beyond me - at least, without digging into the formal proof - but my understanding is that we can prove that pi cannot be represented as a ratio of two integers, and therefore cannot have a finite decimal representation.


We know all these numbers are irrational. That's been known for a long while. Normal numbers are something different, however.

You can look at the wikipedia definition[1], but that involves a few levels of definitions that I don't know, like density, but the gist is that every sequence of N digits occurs with equal frequency to every other sequence of N digits in the expansion of the number.

The definition is complicated due to dealing with infinity and multiple bases.

But that said even with this superficial understanding we can see two things: * Rational numbers can never be normal, since the digits repeat after some period. (Just choose a sequence of numbers longer than the period and you can easily construct a sequence that doesn't appear) * Normal numbers contain every sequence of N digits in their decimal expansion. So if we prove pi is normal (like we believe it is) then we know somewhere in its decimal expansion we can find any sequence of digits we want. Which is the property this comment[2] was referring to.

[1]: https://en.wikipedia.org/wiki/Normal_number [2]: https://news.ycombinator.com/item?id=25282609


Hey was this a serious comment or was it meant to be tongue in cheek? thaumasiotes and I can’t seem to figure it out.


> "Don't choose digits from your favorite irrational number"

What about numbers of which we don't know if they are irrational or not? Like e+π, e⋅π or 2^e.



There's a fallacy in here somewhere. Let's say you pick a common sequence, and you win but have to share the winnings with several people. You say "Darn, I should have picked different numbers!" But then you would have won nothing.


Before the draw every set of numbers has the same probability of coming up. Your expected win differs on the uniqueness of the numbers you choose though, so you might as well choose unique ones.

Yes, of course you can say ex post that you should have picked the numbers that won, but that's not really useful.


Your comment implies you know which numbers are likely to be drawn, which you don't. When every number has an equal chance of being drawn and you don't know what is going to be drawn, it makes more sense to pick numbers that are less popular and thus you're less likely to share the winnings.


Also,instead of buying $10 in a weekly lotto every week, you save your money and wait for it to cross a threshold of value and then buy using the saved money.

50 entries into a $70 million lotto is better than 1 entry, 50 times onto an average lotto of $2 million. The expected payout of vastly bigger.

I've done the math on my local lotteries and I only buy tickets when the EV of the return is greater than the cost of a single ticket. Even then it's only a small amount of tickets. The odds may be in my favour but they're still really small to win.


> I've done the math on my local lotteries and I only buy tickets when the EV of the return is greater than the cost of a single ticket.

If you want to get really technical, you also need to consider the number of people playing. When the jackpot gets extra high more people tend to play, so the likelihood of having to split a jackpot (which happens all the time) becomes higher. Without doing the math I would guess EV of the return is rarely ever greater than the cost of a single ticket.


I factored that in when I did that calculations. How often are their multiple winners and when there are multiple, how many. A surprising amount of the return comes from the non-jackpot prizes, 58% or something. You'll see crowds lining up to buy tickets and yet no winner for a few weeks.


What the maths actually says is your $10 is more likely to make money if you don't do the lotto at all.

If you want to make money gambling then you need a game where you play against other people (like poker) rather than against the house because the house will always stack the odds in its favour.


> EV of the return is greater than the cost of a single ticket.

Wait, if your math is correct, why don't you bet the house on this? With a large enough purchase you're guaranteed to make more money than you spend.


The thing is, his maths isn't correct. Once upon a time some lottery jackpots would occasionally exceed the cost of buying every ticket. But then crime rings cottoned on to that and bought thousands of tickets. Now lotteries take that probability into account and cap the highest rollover amount. Some even have schemes where if there isn't a winning ticket the prize jackpot trickles down to the next highest tickets -- which is a great way to sell massively more tickets and keep the jackpot low enough that it doesn't exceed the ticket price.


Because I don't have the $50 million laying around to do it. The EV is only in my favour when I factor in the small prizes PLUS odds of main jackpot / value of main jackpot.


It doesn't take into account splitting the jackpot. Maybe the expected value is positive if you're the sole winner, but not when 3 other people picked the same numbers as you.


This is the reason why, rarely when I play, I avoid the quickpick and struggle to pick random numbers on the spot. They usually are in increasing order so I am aware I am a bad random number picker. Still, I’d rather share some win than nothing, so in the end it doesn’t make much of a difference


Quick picks are quite random and, from my experience, I'd argue that over 2/3rds would be different from how humans pick numbers.


You might be right, but until i see the algorithm they use I’ll continue to have my own reservations. Their incentive to issue similar numbers and if not hit, the chances to increase the jackpot make a whole lot more people pour money into the game. The house always wins regardless of what numbers come out but if jackpot goes up the pool grows exponentially and so does their profit


Funnily, my math teacher always said that if she would play the lottery, she would just play the number 1, 2, 3, 4, etc. as nobody plays them and they have equal chances of being picked. I guess she was wrong.


How can a math teacher make such an obvious mistake? She's right it's picked less often than all other sequences combined, but that's not the same as being picked less often than every other sequence individually.

Best lottery advice I heard from a math teacher was only make bets that have a chance at a retire-early sized prize and preferably no chance of small winnings. The idea being that when he loses, he's donating to charity, which he would have done anyway.


Wait what? Would have donated to charity anyway... except he didn't donate to any charity, he played the lottery instead.


In most of the US lottery proceeds go to schools. Not sure about other areas.

Of course, the fallacy here is that dollars in budgets are fungible, and when lotteries are established municipalities often redirect the same number of dollars away from the schools into whatever pork barrel projects they like.

That way you have your cake (technically the lottery money does fund schools) and can eat it too (in reality it funds other stuff under the table).

https://www.youtube.com/watch?v=9PK-netuhHA&t=650


> In most of the US lottery proceeds go to schools.

I'm well aware. A government-run school system, however, is NOT a charity. And it's an enormously inefficient way to contribute to a cause -- more than half of the teacher's "donation" is kept by the lottery (distributed as prizes, vendor fees, admin costs)

Nor does much of the money actually impact kids. Most lottery-based education funding in the US is either misleading, or simply replacing (rather than adding to) other funding sources. For example, in New York:

https://www.wgrz.com/article/news/education/how-much-lottery...

“People think the money is going strictly for education, like for books, or schools, or to pay teacher salaries, but it’s not,” DiPietro told 2 On Your Side.

According to DiPietro, the money on occasion has been “pinched off” by the state, to pay for a variety of items, including attorney’s fees for construction projects and even to pave roads near schools.

“They could say there are school busses that are going to drive on this road so the spending would be ‘education based’ when it’s really not, to me that’s a stretch,” he said.


Oh yeah, the creative reallocation of school money is a problem. But this is the society we live in now; a significant part of the population believes that public schooling is a charity.


Choose higher numbers as people tend to choose low ones

Also there's sites for picking numbers that others didn't but you still share the pot.


What - so I choose 1,2,3,4,5 and you choose 6,7,8,9,10 and this we both double our chances for half the pot ?

I can imagine that just leads to a lot of litigation



> Don't choose any pattern of numbers

...unless that pattern of numbers wins...


You mean all this time I should have been picking the winning numbers? Why didn't I think of that?


Better to win a partial pot than none at all, yes?


But if two numbers have an equal chance of being drawn, it's better to choose a pot you won't have to share.


But the suggestion in the article, reasonable in my opinion, is that this sequence was in fact favored for some nefarious reason.


The drawing of these numbers is under question, but you would expect a significant proportion of players to select sequences such as this, even if it's not in their best interests to do so.


Right, which IMO makes it less likely that someone is cheating. If you are going to cheat, why share?


you have to share. A single person couldn't pull this off. A bunch of lottery officials got together and picked these numbers. They had to pick an obvious sequence, because it would not be credible for 20 people to pick a non-obvious sequence.

That's just my theory, of course.


But why would they choose this incredibly specific set of numbers? If you can influence the draw, why influence it in the only way that riles up every human looking at the result? Why not just have it pick something that looks random?


Any combination of numbers has the same probability of showing up in a fair lottery. Why would you choose something like 1 2 3 4 5... that has basically same chance of happening but almost guarantees you will have to share the pot with a bunch of other people?


yes, but that's not the right framing of the problem.

if all outcomes are equally likely from the machine, but humans are more likely to choose numbers with personal significance.

Therefore more tickets will be sold with 1-31 in them vs. higher numbers, and the expected value of your ticket will therefore be higher if you include numbers above them.


Regardless of how you choose your number sequence, buy two of the same sequence. If it's a split, you get twice the share.


No, you don't. If it's a split with one other person, you get 2/3 instead of 1/2, that's only 33% more. If it's a split with two other people, that's 2/4 instead of 1/3, which is only 50% more. etc. and always less than twice as much.

And in all the other cases where it's not a split (either you lose or you win alone), you paid twice as much.


No you see, as the number of winners approaches infinity, the share does double. And goes to zero. So uh.


Of course, you are correct.


From the point of view of your investment (ie. a single ticket) you are guaranteed you will have to split (with the other ticket).

You would do much better by buying two tickets with independent sequences, which is going to double your chance of winning.


This really lowers your expected value by a lot What you describe is very very far from the optimal strategy.


Here in South Africa we have plenty of reasons to investigate the lottopreneurs looting the corrupt lottery: https://groundup.org.za/article/hawks-set-special-team-inves...

But I can assure you this is not fraud, because our fraudsters aren't dumb enough to pick sequential numbers and draw even more attention to themselves.


Hoe gaan dit, Petrus?

> our fraudsters aren't dumb enough to pick...

Maybe they are bored of getting away with it? I think the main other necessary data point is that this was an "electronic" lottery. There was a video somewhere (can't find it) where the National Lottery supposedly explains why "this isn't unusual".

In the video they must have mentioned the phrase "random number generator" at least 5 times and never explained why the numbers aren't unusual.

Brings to mind the people in the 2000s that built dice throwing machines and then piped the results to online random number services. Whatever was wrong with our classic lottery ball machine?

PJvZ


"That's the same combination I have on my luggage!" [0]

0: https://youtu.be/a6iW-8xPw3k


not to mention that they are sharing the prize with 19 other people. Would be much better to pick numbers that others are unlikely to have if you are trying to run a scam.


Hmmmm given that a few years ago Zuma had a pool built at his house on the taxpayers dime and tried to say it was a fire pool nothing would surprise me.


> because our fraudsters aren't dumb enough to pick sequential numbers

That is quite a claim you make there!


While every outcome is equally likely, one can say that, a-priori, the odds of an easily described outcome is small. This is formalized in Algorithmic Information Theory [1] [2].

For instance, when flipping a coin 100 times, the odds of the outcome being describable by a 40-bit program in a predetermined language are only 1 in 2^60.

[1] https://en.wikipedia.org/wiki/Algorithmic_information_theory

[2] https://en.wikipedia.org/wiki/Algorithmically_random_sequenc...


Yes, good point. It's also formalized in basic probability theory. The set of number sequences a human would find interesting or surprising is smaller than the set of all number sequences. If all number sequences are equally likely, by the third axiom of probability you're more likely to draw an uninteresting number sequence than an interesting number sequence.

However that does change the problem statement; we do still expect "123456" to occur equally as often as any other sequence, interesting or not.


It's reckoned in the UK that about 10,000 people every week choose 1, 2, 3, 4, 5, 6 as their lottery numbers. So it'd be really stupid to enter the lottery with those as your own numbers because you'd only win 1/10,000th of the jackpot. In this story 20 people shared the jackpot because of a common sequence.

This seems to indicate that if you are going to enter, you should choose high numbers that aren't, for example, part of memorable sequences, dates, etc. So all numbers > 31. Avoid all even or all odd, progressions, etc.


The best rule to follow to avoid getting duplicate numbers is to just not pick them at all. Quick Pick is a common lottery feature that just gives you random numbers, and should give the best odds of choosing a number which no one else has. Any system your brain can come up with to choose “random” numbers is probably one that someone else came up with too!


It won't give you the best odds.

A trivially constructed strategy with higher expected value: A surprising number of people choose 1, 2, 3, 4, 5, 6 as their numbers. So, instead of a strategy of using Quick Pick, you could use a strategy of starting with Quick Pick and discarding the result until it's something other than 1, 2, 3, 4, 5, 6, which you then use. That has a higher expected value than Quick Pick by itself. The number of people using this strategy is dwarfed by the number of people using the "1, 2, 3, 4, 5, 6" strategy.

This could be expanded to avoiding months, birthdays, etc.


The odds of a quick pick coming up as 1, 2, 3, 4, 5, 6 are roughly equivalent to winning the lottery in the first place, so “repeat the quick pick until you don’t get this particular combination” seems rather silly, no?


I think you're missing the point. The sequence 1, 2, 3, 4, 5, 6 is just one of many common sequences people play. If you have a primitive bool IsCommonSequence(sequence), then you can do better than random by re-rolling whenever IsCommonSequence returns true. As illustrated by some of the other posts on the thread, your odds of getting a common sequence aren't totally astronomical


That’s not what scarmig said though:

> you could use a strategy of starting with Quick Pick and discarding the result until it's something other than 1, 2, 3, 4, 5, 6, which you then use


Yes, this is a nice addition that will definitely cause a significant (thought very small magnitude) shift towards higher profits. Thanks!


That's the HN I know and love.


Yep, the same concept applies to Daily Fantasy Sports. Where you don't necessarily want to pick the best players. You want to pick the best players that no one else will pick. This is for the DFS games that have one player take all.


True but... these people actually won the lottery, what have you won?


I'm much richer because I've never played the lottery.


I do not understand your comment.


exactly. 1/20th of a lottery is still a win.


The numbers weren't drawn out in sequence, but rather they're arranged in sequence after the draw (makes it easier to check). The article also didn't mention this, but a random-number generator is used. The older physical draw used to be audited, but I'm not sure if what happens now.

There are a lot of shady things being done with the profits of the lottery, like non-existent charities being funded, it's basically a fund that gets looted.

Even with that, the probability of this combination is 1/42 million. The draw has 5/50 and 1/20 powerballs.

The best thing about this happening is that 99 people got the 5-9 numbers correct, so this was a good redistribution of funds. The prize was huge (114m), so each winner for a sizeable chunk.

Was it rigged? I doubt it, but I welcome the probe, as there is corruption in the SA lottery (even down to the awarding of the contract to run it).

I'm on mobile, so I didn't share links, of one would like them, I can add them.


Dilbert comic strip: https://dilbert.com/strip/2001-10-25

  "Over here we have our random number generator."
  "Nine nine nine nine nine nine"
  "Are you sure that's random?"
  "Thats the problem with randomness: you can never be sure."


Randomness is a weird concept, because if you roll a die it's random right? Unless you have all the physical variables - roll speed, angular velocity, height, shape of die, air resistance, air movement, temperature, etc to work out where it will land. So randomness is more a lack of knowledge. Randomness is that you are not sure.

And I don't know enough about quantum mechanics to even comment on that, but that's some weird shit too!


There was an interesting discussion about this the other day: https://news.ycombinator.com/item?id=25237356


That is my absolute favorite Dilbert.



Similar-ish story from Bulgaria back in 2009.

4, 15, 23, 24, 35, 42 were drawn back to back weeks live on TV. They also had a probe but nothing malicious have been found. RNG being RNG.

http://news.bbc.co.uk/2/hi/8259801.stm


The probability of that happening is much much negligible though (the probability of it happening once is 1/N, the probability of it happening twice is 1/N^2)


That only applies if those particular numbers are somehow significant. The chance of picking last week's numbers are the same as picking any other combination.


RNG doesn't do that. The probability is infinitesimal. It didn't happen by accident. Its fixed.

I can say that with something like 99.99999% confidence. Which doesn't mean "See it could happen!". It means "No its never gonna happen"


So because the chances of something happening are so infinitesimally small, it’ll never happen? Airplanes are orders of magnitude safer than cars, yet planes still crash. The “law of large numbers” and all.


But there are simpler explanations, like they broadcast last weeks drawing by mistake, and once broadcast, they can't undo it.


FTA:

> The lottery organisers described it as a freak coincidence and pointed out that the numbers were drawn in a different order.

So, not the same broadcast.


Yes, if you get small enough (not saying we are neccesarily small enough here).

If the probability was something like 1/2^512, and it happened, i think its safe to say that something is fixed.


If you shuffle a deck of cards, the probability of the cards ending up in the order you got is 1 in 8e67, but there it is.


There's also a very small probability that the atoms inside the cards will spontanously shift so that every single card is an ace of spades


I don't know how often the Bulgarian lottery runs, and how many number there are, but I expect the probability of it happening during the life of the lottery would be about 1 in thousands. That's low, but not unbelievably so.

We need to consider all the "interesting" things that can happen. It can be sequential numbers like in the article, or anything that strikes you when it happens but didn't think about before.

There is still a chance for a mistake. Fraud is unlikely: why would a fraudster do that? It is extremely noticeable and is likely to make the earnings shared.


The probability is roughly one in a million. The problem is that people underestimate how often one-in-a-million chances happen, because they don't think about how many times you actually roll for that one-in-a-million chance.

Take the Bulgaria example. How many lotteries are run around the world, such that we'd hear about a rare coincidence like this? Probably about a 1000 a week. So now the one-in-a-million chance is a one-in-a-thousand chance on a weekly basis, or put differently, we should expect to see it happen about every 20 years.


Since there are 50 balls and 20 powerballs, its roughly 1 in 20 million. So, do the draw 20 million times to get a particular sequence.


That's not how probability works. Each draw is an independent event from every previous draw, doing a draw with a 1 in 20 million chance of a given sequence 20 million times will not draw every sequence once.


37% chance you won't get it, 37% chance you'll get it exactly once.


Probability tricks us. I'm not an expert but this is what I think.

Follow one lottery, and then watch it this weekend. If the same numbers come up this weekend as last - it is probably fixed.

Now follow every lottery, and watch them all over a few years. If the same numbers come up one week as last, it could be fix (maybe higher probability than normal) but it's most likely a fluke.


The probability is exactly the same as the probability of any one ticket holder winning the lottery.

It's a very simple argument: the draw from the week before is the ticket. The next draw, which is completely independent, either matches the ticket, or it doesn't.


That's how the infinite improbability drive works.

https://www.youtube.com/watch?v=UIUIO1ImtJo


I've told the few people I know who play the lotto they should pick "1,2,3,4,5,6" and their faces screw up. I tell them it's just as likely. They don't believe me. I ask, then why are you even playing if you don't understand the game? That doesn't go over well.


I used to do the same. Told my mom to buy that ticket, and explained to her she was wrong. And that made her sad to realize.

She buys the tickets because she enjoys the pleasure of forward predictions: what will she do with the money. Go to the bahamas, buy a villa. Relax.

That forward projection means dopamine and norephedrine release -- similar to taking a drug. She enjoys that pleasure that's why she buys it.


Hear hear. That's exactly why I occasionally buy a lotto ticket or 20 when the number gets REALLY big. I consider $20 every year toward an hour of fantasizing about what I'd do with a $500M lump sum a worthwhile investment.

($500M lump sum...$250M after taxes...invested into triple tax-free muni bonds paying 2.5%....ahhhh...$521,000 a month for life...mmmmmm)


Norepinephrine?


It may be just as likely to win but it's not just as good, the expected value of the prize is much smaller because the prize will be shared with a large number of people. Better to choose a random set of numbers.


But that's a completely different point.


It isn't because if you play every week you should be looking at EV and returns - not the likelihood of winning the entire jackpot on one run

With prize pools often paying down to a few numbers selected you could be +EV over the field if you pick numbers they don't


I don't play.

The point was, the people I talk to who do, don't understand probability. Nothing else matters until they understand probability.


But it doesn't seem like you have a very good understanding of EV and probability either since you recommend a completely sub-optimal strategy.


I'm not trying to help them win. I'd rather they not play at all. It's throwing money away.


He's not suggesting a strategy, he's highlighting a controversial pick to get them to question their assumptions.


You're stating it like only lottery players don't understand probabilities. Read the top comments in the thread, plenty of educated, intelligent HN readers have no clue how to calculate these probabilities either.

In fact, most lottery players I've met DO understand probability, or at least grasp the fact that their "investment" is, in the long term, an expensive hobby.


> You're stating it like only lottery players don't understand probabilities.

OP doesn't say that literally, and I'm not reading it that way, either.

> In fact, most lottery players I've met DO understand probability

That's great! However, OP might be talking to and about different people here. People who don't know probabilities well exist, and some of them play the lottery. In fact, I think there's a lot of them.

I think all OP is saying that it's better for people to understand probability better, and then decide to play the lottery, like the people that you're talking to.


But they seem to have had an intuitive grasp for something out of your reach. Seems like a wash on superiority complexes.


You're making some assumptions here about people, their conversations and motives, all of which you don't know, to come to some sweeping negative conclusion.

I have no trouble at all believing that guy knows some people who have a bad grasp on probabilities, especially if they also happen to play the lottery. ;)


Yes, but it's also a good point!


As we see from this very article, while 1,2,3,4,5,6 has the same chance of being drawn it has a higher chance of being called a scam. So lower chance that you will see money wired to your account.


Lottery should be illegal. It is a luck-based system to concentrate and redistribute wealth from individuals (often lower class) to a single member of the society; propelling one person from the lower echelons of the society to the top at the expense of others. We want to align some kind of a productive output and reasonable incentives for a person to justify amassing so much wealth.


That is true only if making lotteries illegal made them nonexistent. In the United States, the legal lottery is arguably better than the widespread (illegal) "numbers game" it replaced.


It's also arguably worse. It adds a conflict of interest that the "numbers game" never had. The same government that is responsible for providing an education that makes people resistant to the lottery has an interest in people being vulnerable to the lottery.


In my home state of Florida, it's pretty nefarious because they often advertise that the proceeds are used to support education. While that is true, they also removed other sources of funds such as taxes. So in the end the funding level is about the same while less of it comes from taxes. This means Florida can tax people less to pay for education but as other have pointed out, the lottery is essentially a tax on stupidity or at least innumeracy. I guess using an innumeracy tax to fund education is kind of poetic but what Florida has done is tax the uneducated to give tax cuts to everyone else.


I don't believe that lottery players are unintelligent or uneducated.

It's rational to play the lottery if you believe that it's the only way to make a significant economic change to your life.

That's a bigger problem than just education provision.


Why? It’s just gambling!

Last year the HN consensus was that daily fantasy sports betting was some sort of civil liberty.


The lack of a path to changing your economic circumstances through work or education is the problem.


I'd be good if all money made through the state sponsored lottery would go back into funding for education on gambling


Illegal numbers would never reach 5% of the revenue of legal state lotteries. Probably not even 1%. Plenty of law-abiding people with little or no connection to any underworld activity are nevertheless bombarded with legal state-run lottery ads out in the open and available to play at the grocery or convenience store they visit frequently, even daily -- illegal games wouldn't have ads, wouldn't be at 7-11, drawings wouldn't be televised. Further, all numbers games are low-stakes... there is no crime syndicate which could ever afford to run a Powerball or MegaMillions game with 9-figure prizes. Even 6-digit prizes wouldn't be affordable, even if it was, who would ever trust an underground game with that kind of money?


Which is why in the US at least, most states take about 40-50% of the lottery winnings to fund education programs. In Georgia, we have the HOPE scholarship which is funded by the Georgia Lottery. If you have a good GPA in high school, and you maintain it in college, you basically get a free ride and don't have to pay tuition. While the HOPE scholarship can probably be tailored, more towards the household income, from a distribution perspective, it still works well.

It allowed myself and many of my peers to graduate with significantly less debt than if we had to pay for college ourselves.


> most states take about 40-50% of the lottery winnings to fund education programs. In Georgia, we have the HOPE scholarship which is funded by the Georgia Lottery.

My father and I once debated the merits of this, and he made a point that stuck with me: the overall funding for education didn't change. It's just that the money that used to come from the government now comes from lotteries instead.

Meaning the burden of that cost has shifted from the collected taxes, which is more proportionally paid by the rich, to the lottery system which is more proportionally paid by the poor.


Did the amount of money the poor invest in gambling change? Or are the gambling losses just being re-targeted to fund education, instead whatever they used to end up in previously.


That's a fun point!

I am 100% going to bring that up with the old man the next time we're allowed in the same room[0]. I think he'll like it.

[0] Covid risks...


In a fair number of states, its funny math. They don't increase the size of the budget, they just shift the money around. So if lottery revenue goes up for the state, more of the education budget may come from lottery money vs taxes, but the overall budget remains the same.


Exactly. "Please vote for our 1% sales tax increase! It's all going to education, we swear! Think of the children!"

Well sure, you're allocating the new dollars to education, then moving the old allocations over to your pet project...


States do this when they pass gambling laws to make it seem less scummy in my opinion.

My state recently made certain slot machine games legal. I found myself overlooking the parking lot of one of the casinos at 8am while waiting for my grocery pickup. The addicts were already arriving. Somehow every car seemed to have been in an accident and not been repaired. The parking lot was covered in oil stains.

When the casinos opened I imagined people going out occasionally for a fun night gambling. This was far from that.


The trouble with earmarking like this is that the legislature takes it into account when distributing funds, so you can easily have a situation where "all proceeds from the lottery go to education" but you don't actually have any more education funding than you did before.

A specific, new, fully funded program like a scholarship is actually possibly a better form of this than just general funding for the school system.


Funding the public education system is a good thing and a similar system exists in California: https://www.calottery.com/who-benefits

However OPs point about mostly lower socioeconomic groups funding the lottery is still relevant as it raises questions about whether this pseudo-regressive “tax” is a fair way to fund a public good like education.


In all fairness, in the US--at least where I live--the bulk of education funding comes from property taxes which is presumably progressive. (There is a different criticism that this leads to wealthier towns having better education. Which is probably true at some level although there isn't a lot of correlation between per pupil spend and educational outcomes in a given area.)


Property taxes aren't progressive: although they're levied on individuals who tend to be well-off, a significant proportion of those costs end up passed on to renters. Additionally, poorer people tend to spend a larger portion of their income on housing (be it rent or otherwise) than the rich.


That's fair enough and probably depends on the housing prices in a location. Wealthier people tend to own more house but it may still be a smaller portion of their income, depending upon where they live.


HOPE scholarships are great PR but awful policy. Basically, the poorest people in Georgia are paying for the children of mostly middle and upper-class residents to go to UGA and Ga Tech for free. Of course the state points out the handful of low-income students who also benefited from a HOPE scholarship, but these are the minority of recipients, and they would almost all have qualified for significant grants and financial aid anyway. The most offensive aspect, I'll state again, is the fact that the poorest residents overwhelmingly purchase lottery tickets while the wealthiest residents overwhelmingly receive the scholarships.

It also makes the state university (UGA) much, much harder to gain admission to, since demand for a free college education is high. Applications (and thus necessary GPA/SAT scores) are through the roof... Ivy-qualified students (again, often from wealthy families) are often encouraged to go to UGA for free instead.

HOPE already vastly exceeded its budget once, leading to cutbacks and reforms about a decade ago, with many students losing scholarships or suddenly not qualifiying. A more recent report says it is likely to run out of funds again in 2028. https://www.11alive.com/article/news/local/new-analysis-show...


True, but it still amounts to a very regressive tax, even if the proceeds are fed back into education.


The cynic in me notes that the stock exchange amounts to where moderate to middle class does all of their gambling. With most of the proceeds just consolidating up the wealth chain. At least in lotteries the programs are progressive. :D


That's different. The average "gambler" in the stock exchange comes out ahead.


If you buy and hold stocks for a long time it's not the same thing as betting on a horse or buying a lottery ticket.


Which is used to argue for reduced progressive (or even flat) tax-based support for schools.


It's a high risk and very high variance bet. The odds are terrible but for 99.999% of people it's the only bet you can make with "instantly project me into the top echelon of wealth" levels of upside. There's value in that which is not captured by the negative expected value.

Finance professionals have access to this kind of bet outside the lottery via leverage and complex derivatives and they place bets all the time. Earlier this year Bill Ackman made 2.6B on a really wacky bet against corporate debt [0].

The lottery is absolutely a waste of money for most people, but if you're rational about it you can see it as an opportunity that has no replacement.

0: https://www.cbsnews.com/news/bill-ackman-billionaire-made-2-...


To put lottery winnings in to context, the biggest ever lottery win in the US was $1.56bn, or approximately 0.84% of Jeff Bezos's current wealth.


I more think its a system to tax the poor without them being mad about it.


> We want to align some kind of a productive output and reasonable incentives for a person to justify amassing so much wealth.

Or maybe it's just fun to play, and people can make their own choices?

Frankly, I think governments should not be in the business of attempting to engineer "desirable" outcomes through coercion. Invest the money that would have been spent on coercion/crackdowns on education and then let people live with the consequences of their actions.


A lot of fun things are fundamentally unproductive, risk taking and surprises that playing the lottery entail are fun to a lot of people.


> Lottery should be illegal.

Fortunately, we can always rely on crime syndicates to provide unrestricted access to vices (either gambling, sex, booze, or drugs) for as long as people will want it, no matter what Rulers and other "pezzonovante" think of such a vices.

Said otherwise, you can not change, and will not change, human nature.


It would seem to me that all wealth is at the expense of others. I like your comment, but I think it needs to account for the fact that non-lottery wealth comes from either being lucky (born into it, or first to market) or by taking from other people.


That seems a little too pessimistic to me. The economy is absolutely not a zero sum game.


Looked at individual transactions within the economy - i.e. how this thread is looking at the lottery - those do appear to be zero sum in the same way.


It's usually other people voluntarily give it to them, not that they take it.

First to market isn't just luck. It's often the result of work harder and faster and taking more risk than others. It's not as if Bill Gates just sat around and told people to make software.


While I agree, I very much enjoy gambling because it is fun, and gets me the dopamine hit I'm after :)


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