South Africa's lottery probed as 5, 6, 7, 8, 9 and 10 drawn
bbc.co.uk
bbc.co.uk
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.)
https://www.wolframalpha.com/input/?i=%281-1%2F%2850choose5%...
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.
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?
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)).
But the main point was the methodology. (1 - (1 - chance_of_sequence)^(n_draws)).
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?
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.”
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".
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.
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.
2*(1/49)*(1/48)*(1/47)*(1/46)*365*1000*(45/50) = 12.9%
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%
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.
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
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.
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.
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. ;)
There’s still some chance it doesn’t happen in one year.
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.
It's been clear throughout this thread what was meant.
Probably something like reciprocal half-life; 775% / yr is about 1 / 1.5 month ie, happens about once every month and a half.
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.
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?
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...
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.
(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.
https://www.gamblingsites.org/blog/how-many-different-lotter...
Looks like a reasonable analysis. Their total is 180.
You have: 50^6 / (50*2) / (1000/day)
You want: yr
* 427.79832
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).
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.
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.
This might not be in the news because it's so rare. It might just be a slow news day.
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?
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.
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.
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 ;)
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.
https://www.wolframalpha.com/input/?i=binomial+calculator&as...
If they are 200 draws per week, the probability is 26% over 10 years.
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.
The list of squares looks more like powers of two btw ;)
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.
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.
EDIT: Oh, the best part: we have this "lottery for the impatient" that has draws every 4 minutes :)
In any case, there's probably a few people out there who picked "1 2 3 4 5" that are kicking themselves extra hard.
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.
[0]: https://en.wikipedia.org/wiki/Interesting_number_paradox
(Note that Kolmogorov complexity is not generally computable, because you could solve the halting problem if you can compute it for all sequences).
In general the probability that some unlikely events occur is high.
(Code here: <https://gist.github.com/wolfgang42/2df001b05065488620700f0fd...>)
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 :-)
[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.
aka sad stories of the mentally ill people your company was exploiting.
- Ambrose Bierce, The Unabridged Devil's Dictionary
If I was completely rational I'd bet my house on it when it tips into my favour. 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.
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.
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.
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.
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.
That's just my theory, of course.
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.
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".
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.
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.)
- 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.
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.
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.
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
All normal numbers have the property you mentioned, and nearly all irrational numbers are normal, but there are some that are not.
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.
This is not a universal convention.
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
What about numbers of which we don't know if they are irrational or not? Like e+π, e⋅π or 2^e.
Also there's sites for picking numbers that others didn't but you still share the pot.
I can imagine that just leads to a lot of litigation
...unless that pattern of numbers wins...
And in all the other cases where it's not a split (either you lose or you win alone), you paid twice as much.
You would do much better by buying two tickets with independent sequences, which is going to double your chance of winning.
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.
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).
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.
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.
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.
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.
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.
Yes, of course you can say ex post that you should have picked the numbers that won, but that's not really useful.
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.
> 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 is quite a claim you make there!
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...
However that does change the problem statement; we do still expect "123456" to occur equally as often as any other sequence, interesting or not.
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.
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.
> 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
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.
"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."And I don't know enough about quantum mechanics to even comment on that, but that's some weird shit too!
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.
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"
> 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.
If the probability was something like 1/2^512, and it happened, i think its safe to say that something is fixed.
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.
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.
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.
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.
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.
($500M lump sum...$250M after taxes...invested into triple tax-free muni bonds paying 2.5%....ahhhh...$521,000 a month for life...mmmmmm)
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
The point was, the people I talk to who do, don't understand probability. Nothing else matters until they understand probability.
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. ;)
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.
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.
It allowed myself and many of my peers to graduate with significantly less debt than if we had to pay for college ourselves.
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.
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.
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.
Well sure, you're allocating the new dollars to education, then moving the old allocations over to your pet project...
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.
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...
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...
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.
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.
Last year the HN consensus was that daily fantasy sports betting was some sort of civil liberty.
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.
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-...
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.
Why on earth would you pick numbers that were certain to draw attention?
edit: wait for morse code S-O-S :)
There’s a paradox related to this train of thought: the “unexpected hanging paradox.”[0]
[0]: https://en.wikipedia.org/wiki/Unexpected_hanging_paradox
Not only that, but sequences like this are significantly more likely to have many winners (assuming the lottery allows ticket purchasers to select their own numbers).
You are no less likely to win, but you are more likely to split any winnings with another player who also bought that number, lowering the expected value of the ticket.
If it produces pretty sequences at significantly greater than expected rates, that would be suspicious.
Absolutely astonishing.
As an aside, the inevitability of weird flukes in large sample sizes really sinks in if you've ever wasted part of a year multi-tabling online poker. You'll see some hands play out in totally WTF fashion, as 1:100 and 1:1,000 probabilities come good.
The lesson, of course, is that when you play thousands of hands, encountering such oddities with your money at stake is no longer nearly impossible. It's moved into the zone of nearly certain.
Such a sequence, assuming a low total number of draws, is pretty strong evidence in favor of the theory that the balls were put in sequentially and were supposed to be randomized but the randomization didn't actually happen.
Public faith is far more valuable than any cost that could be incurred by an otherwise useless investigation into math.
https://www.bbc.co.uk/news/world-africa-55154525
Would be just a lovely Easter egg if someone intervened and updated it to be "world-africa-5678910"
...for the curious.
Because the scene is priceless
So lost I was. Never heard of Spaceballs.
But what I imagine it COULD be, is that they have some kind of system that can deterministically generate a specific sequence of numbers. So once in a while they can generate a sequence that'll reward someone they know (or every time, but that'd probably be discovered quickly). Someone left that system in a test mode which generates the "5...10" sequence to demonstrate that the system works.
But I think it's still more likely that the odds of a lottery somewhere in the world has some kind of interesting sequence is surpisingly high, as others here have said.
If you choose (which you shouldn't) to play the lottery, then avoid common sequences like 5,6,7,8,9,10 or dates. In very unlikely case that you win, you'll most probably have to share the prize
[0] https://en.wikipedia.org/wiki/Fano_plane [1] https://en.wikipedia.org/wiki/Transylvania_lottery
The only good reason to investigate this is politics. That is to say, if six consecutive integers come up, and the lottery corporation doesn't make some kind of public show of investigating it, they risk inviting criticism from ignoramuses.
> the PowerBall jackpot draw require[s] players to pick five main numbers from 1 to 45 and one 'PowerBall' number from 1 to 20 for an entry fee of R5 per board
As suggested elsewhere here, this could be someone who has broken the system trying to draw attention to the flaw. If that's the case, why choose this sequence then? Why not go for 1, 2, 3, 4, 5, 6?
https://www.turkishminute.com/2020/11/27/turkish-mp-hints-at...
The accents maybe a bit hard to understand.
Youtube auto closed captions agrees with this statement.
Shortly after the 9/11 terror attacks on New York, the Pick 3 came up with 9-1-1. Math happens.
In Bulgaria the same numbers were once drawn in 2 subsequent weeks.
In Germany the same numbers were drawn as in the Netherlands the week before.
However, the probability of ending up with a sequence like this is still quite low - probability of coming up with six consecutive numbers is 40 : 42,375,200 i.e. less than one in a million.
> The chances of winning South Africa's PowerBall lottery are one in 42,375,200
I don't know how many balls they have and whether there is replacement to tell you the actual number, but there shouldn't be anything special about this one among all the other possible sequences, except that it's more likely to be guessed.
1,2,3,4,5,6 and 10,11,12,13,14,15 and even 2,4,6,8,10,12 would have had the same amount of attention and scruitny.
So "what are the odds that a sequence of numbers would come out that would look 'funny' to many people" is now not one out of n, it's several hundred out of n.
This can be seen in the official announcement: https://youtu.be/11RN1pEnrTo
To give you a more charitable interpretation, the South African lottery Powerball can be 1-20 so there are 20 straights possible so 20 in 42,375,200. Or roughly 1 in 2,000,000 or roughly 40,000 years of weekly draws to see a straight.
There is some inherent fuzziness in what you consider a "suspicious sequence", obviously. If you stick to entire sequences you don't get very many; however, if you consider "1 2 3 4 5" in the original numbers to be suspicious regardless of the bonus ball draw, then the number of such sequences starts going up fairly quickly. (Include things like [1,3,5,7,9], [2,4,6,8,10], [3,6,9,12,15], etc.)
It's pretty easy to get up to a couple thousand "suspicious sequences". For the sake of roundness and being a bit conservative, let's say there are 423 "suspicious sequences", in which case the odds of hitting one are roughly 1 in 100,000. You may choose to add another order of magnitude of "suspicious sequences" pretty easily, in which case it goes to 1 in 10,000. Given the number of draws done for this sort of lottery (dozens or hundreds a day, I'd guess), it's inevitable that sooner or later one would be hit on lotteries of this size.
Some of the challenge in weighting the number of sequences is they aren't all equally "suspicious". [1,2,3,4,5] is what most people would consider a "dead giveaway" (right or wrong), whereas [1,2,4,8,16,32] would probably bother fewer people. Some people might still find even [1,2,3,4,18,bonus 6] suspicious ("look how 'close' it was to 12345!"). So there's just some intrinsic fuzziness to the answer of how likely this is.
> The organisers say the sequence is often picked. But some have alleged a scam and an investigation is under way.
> It is extremely rare for multiple winners to share the jackpot.
that sequence happens, but multiple winners is rare
so the combination of them is the concern here
Not enough people noticing the part where it says "The organisers say the sequence is often picked."
Often?