7 ms·

What Are the Odds?

145 points by jjgreen 4y ago

unphased 4y ago
This article started off really clear and straightforward, then it dove into stuff that I clearly wasn't going to be able to follow, and upon skipping it I arrived at the explanation which was was plainly obvious from common sense.
- rubatuga 4y ago
  Yeah, that article really took a sudden nosedive
- minihat 4y ago
  >such a number is often dutifully provided to such journalists, who in turn report it as some sort of quantitative demonstration of how remarkable the event was. Apparently the conclusions are not common sense if you are a journalist.
- andreareina 4y ago
  Common sense can mislead, so it's instructive to make it rigorous to see whether it does indeed hold up. Not for everything or even most things, otherwise you'd be analysis paralysis sets in.
  - jiggywiggy 4y ago
    In this case the author was a bit silly. He also used common sense to determine he had no way of determining the probability of a rigged event. And therefore using Bayesian probability looked fancy but didn't help learn anything new. I remember reading first paragraph and thinking wow, bayesian has a way of helping determining the odds of that, but no. Either he wanted to educate or is so involved with statistics that it became his default way of framing problems.
    - lupire 4y ago
      > Either he wanted to educate Yeah, Professor Tao has been known to try that from time to time.
- Rioghasarig 4y ago
  Well that's what mathematicians like to do.
ethanwillis 4y ago
And yet what Mr. Tao misses is his own assumption that the drawing of each number is truly independent.
- FabHK 4y ago
  They’re not independent. If the 9 is drawn, the next ball won’t be the 9.
  - rippercushions 4y ago
    Mr Tao accounts for this, since (55 6) = (55/6)*(54/5)*(53/4)*(52/3)*(51/2)*(50/1)
- Tainnor 4y ago
  Yes, surely fields medalist Prof. Terence Tao must have fundamentally misunderstood basic probability.
fny 4y ago
Most useful takeaway from this article: If you choose to play the lotto, don't use cute numbers. Odds are high that other people will too, and you'll have to split the pot. > Part of the explanation surely lies in the unusually large number (433) of lottery winners > But on the previous draw of the same lottery ... > the unremarkable sequence of numbers {11, 26, 33, 45, 51, 55} were drawn ... > and no tickets ended up claiming the jackpot.
- Semaphor 4y ago
  That is actually commonly recommended for exactly that reason. Often right after recommending to not play the lottery at all ;)
  - katrotz 4y ago
    Remember reading in some book on probability and statistics that one will be better off betting insignificant amounts of money rather than not playing at all.
    - Semaphor 4y ago
      Sounds wrong, that would essentially be saying one should play the cheapest level of all lotteries.
      - mellavora 4y ago
        naw, it is probably based on the assumption that the payout can be life-changing big. risking pocket change to win big is quasi-reasonable, even though it is a loosing bet. Because you don't notice the downside/drawdowns. risking more than pocket change is not reasonable, because it is a loosing bet. NB: I don't play the lottery.
        lupire 4y ago
        Similarly, you should make small high-upside bets bets like "strike up a chat with a as stranger"
sohkamyung 4y ago
An alternative take via The Conversation [1]: > Based on anecdotal evidence from other lotteries, this number may not at all be unusual. We also need to consider the many thousands of similar lotteries drawn around the world each year, almost all of which receive no international press. While such outcomes are highly improbable for any given draw, the huge number of total lotteries means it’s actually quite likely at least one of them will produce a remarkable outcome by chance alone. [1] https://theconversation.com/433-people-win-a-lottery-jackpot-impossible-probability-and-psychology-suggest-its-more-likely-than-youd-think-191946 https://theconversation.com/433-people-win-a-lottery-jackpot...
- AdamH12113 4y ago
  The multiple endpoints fallacy also comes into play here. Instead of multiples of 9, it could have been multiples of another number, or any group of sequential numbers, or sequential even numbers, or prime numbers, or... The odds of a particular remarkable thing happening are very low, but the odds of some remarkable thing happening can be much higher.
tnuc 4y ago
Maybe they just got the numbers from a fortune cookie?
- bananatype 4y ago
idiocrat 4y ago
That reminds me that I finally need to start participating in lotto, just to have a skin in the game. Not participating - zero chance of winning. Participating - 1 in 30 million chance of winning 4m USD.
- tomcatfish 4y ago
  0.000000000000001 chance - cost $0 0.000000033333333 chance - cost $4 Which one comes up on top, expected value wise?
  - Epa095 4y ago
    Your evaluation of the price might not scale linearly with the money amount, making it a fault to focus to much on the expected value. Example 1: you own the mafia 1 million dollar, pay tomorrow or die. You have absolutely no way to get the money before the deadline. You have some money though, and the winner of the lottery gets 1 million. Your expected value(in $) is negative, but the chance of surviving is no longer 0 if you play the lottery. Example 2: you are a normal lower income person. You have no real chance of getting rich, or even comfortable. You can save the 4$, and maybe buy a slightly better TV every 3 years, but it will make no real difference in your life. Winning e.g 2 million on the other hand would lead to a qualitatively different (and better) life. Its not that 2 million is 2millon/4 times better than 4$, it is just incomparable. One leads to a completely different life.
    - dkbrk 4y ago
      You're not wrong in principle that you could have a utility function with positive expected value for playing the lottery, but that's misleading. Generally it is taken that money has diminishing marginal utility. This is even more salient for poor people, for whom the marginal value of $1 at their current wealth is vastly more than the value of $1 at the wealth of winning the lottery; i.e. playing the lottery is even worse, in terms of expected value, when you apply a reasonable utility function. You're correct about your mafia example, but that's highly contrived. I contest that your second example does not show what you claim; that is exactly the sort of person who should not be playing the lottery.
FabHK 4y ago
The beginning of the article is an excellent (though conventional) introduction to Bayes and Bayesian updating, but what I found original and have never seen expressed so clearly were the remarks about dishonest discussion tactics, especially the first one: Formulate a vague hypothesis H1 so broad that it is not extremely unlikely (thus has non-negligible prior, P(H1) > 0), then in the updating step sneakily introduce a much more specific hypothesis H1’ that is far less likely a priori, but that yields a high probability for the evidence P(E|H1’). If the audience doesn’t notice the bait and switch, it’ll come away with the impression that the evidence strongly supports H1/H1’ vis-à-vis H0, when it actually doesn’t. Reminiscent of the Motte and Bailey fallacy. https://en.m.wikipedia.org/wiki/Motte-and-bailey_fallacy https://en.m.wikipedia.org/wiki/Motte-and-bailey_fallacy
- mistermann 4y ago
  It's even more complicated than that though: "dishonest discussion tactics" suggests that the person is doing it with fully conscious awareness and intent, but this is far from a safe assumption - and, it isn't a simple binary "is or is not", and it isn't only the usual suspects who commit this technique.
  - simonh 4y ago
    Maybe so, but ultimately it's up to us to take responsibility for the techniques we use in argumentation. I would agree that it's not generally a good idea to try to use an attack on the rhetorical techniques someone uses in order to win an argument. It's best to politely find a way to challenge their argument without attacking them personally for the way they put their case.
    - mistermann 4y ago
      A very reasonable take...but I wonder if there might be a substantially more optimal approach, that is possible.
- pc86 4y ago
  Maybe it's just my liberal arts education but discussions like this and comments like "hypothesis H1’ ... that yields a high probability for the evidence P(E|H1’)" or "evidence strongly supports H1/H1’ vis-à-vis H0" mean absolutely nothing to me without a concrete, real-world example of what H1, H1', and H0 are.
andirk 4y ago
There have been a few national American lotteries in the past handful of years with a jackpot so big ($1.5+ billion) that it was statistically advantageous to buy every number sequence; the total price of the lottery tickets was less than the jackpot. The problem with this lotto hack was that it was not humanly possible to buy that many tickets. Also, if even 1 other person won, the person would be in the hole some 1/2 a billion $ after splitting the winnings.
- jmharvey 4y ago
  Somewhat counterintuitively, the billion-dollar jackpot lotteries' expected values often shrink as the jackpots grow, because the media circus pulls in a lot of players and increase the likelihood of a split jackpot. It's not that uncommon for a smaller lottery to have a positive expected value. I think the logistics of buying every combination in a pick-6 lottery (on the order of 10 million combinations) would be rather unwieldy, but a pick-5 (on the order of a few hundred thousand combos) could be doable with a small team.
  - lupire 4y ago
    The "broken" lotteries aren't for that kind of jackpot, for the reason you say. The broken lotteries are counting -caeds based: when a fixed priced pool is printed on the cards, and then part of the deck is exposed as losers so the rest of the deck has positive EV.
  - tzs 4y ago
    Virginia lottery around the end of 1991: pick 6 from 1 to 44 giving 7 059 052 possible combinations. The challengers: International Lotto Fund (ILF), an Australian investor group with 2500 investors. ILF did't quite manage to cover all 7.1 million combinations. They only managed 5 million. They fell short because they underestimated the time it would take to buy the tickets, which they got from around 8 grocery and convenience store chains. And no, I'm not some kind of lottery history buff. This is one of those weird bits of trivia one picks up in law school. Specifically in the class I took on transnational tax around 1995, where the taxing of ILF's winnings from this hit some edge cases in US and Australian tax law that were interesting and instructive enough for the case to make the next edition of the textbook.
eyelidlessness 4y ago
The odds of N-1 numbers being “remarkable” diminishes for each number drawn. Unless remarkable can be conjured and retconned for every number combination or every Nth number drawn. Each combination is equally likely before the first number is drawn. And equally likely at the end of the draw. But odds change when you have a partial result that’s already ruled out possibilities for each drawing. You can’t have 9, 18, 27, 36, 45, 54 if you draw an 8. The odds of that are zero.
lqet 4y ago
Comment under the article: In particular, the geometry of the lottery tickets seems important. It is plausible that 55 numbers are set up in a 7 x 8 matrix pattern, with one wildcard (to produce 56 objects). Imagine that it is done as such: *, 1, 2, .. , 7 8, 9, 10, .., 15 16, 17, 18, … , 23 24, 25, 26, 27…, 31 .. then multiples of 9 are the main diagonal. That would explain frequency in an easy way.
- ginko 4y ago
  Looks like that's the case. See: https://primer.com.ph/tips-guides/wp-content/uploads/sites/5/2018/10/lotto-play-card-primer.jpg https://primer.com.ph/tips-guides/wp-content/uploads/sites/5...
jzer0cool 4y ago
Interesting question for this lottery draw. 1 - How about past lottery dates. Is there also a high purchase of these same sequence of numbers? 2 - Who are the 433 winners? Is there an unusual clustering of the winners? Was it mostly purchased by a single entity?
- anonymous344 4y ago
  This is actually the only smart question(s) to ask to find out was it a cheat
- bananatype 4y ago
aaron695 4y ago
apienx 4y ago
ELI5 Bayesian theory. https://youtu.be/HZGCoVF3YvM https://youtu.be/HZGCoVF3YvM
botencat 4y ago
It’s an interesting read but the author could have just used Kolmogorov complexity to explain all this.
mvcalder 4y ago
H''''': The lottery is run by corrupt officials, who want to cover their tracks. In order to do so they want to have many associates win so no single winner is subject to scrutiny. They need to communicate to this diverse group of conspirators the correct numbers to choose in a simple way, say: multiples of 9, or anti-diagonal of ticket. Remark 4: The human mind is an amazing hypothesis generating machine. If it also knows about Bayesian statistics, it is capable of accounting for it. Paranoia is a bitch.
graycat 4y ago
For any six numbers that can exist or be drawn, once they are drawn can find something about them that is improbable. Sooooo, that found something improbable about the numbers, once they have been drawn, means essentially nothing.
- kgwgk 4y ago
  On the other hand, those numbers were "special" before they were drawn - which is why more people played those numbers than would be expected for "non-special" combinations.
- lupire 4y ago
  "Essentially" is load-bearing there. You can compute the statistics to determine what number sets are more suspicious.
  - graycat 4y ago
    I don't think we are on the same page: There have to be some uncountably infinitely many statistical tests can apply to six numbers, and given any list of 6 numbers it will have to fail, be very "suspicious", according to some of the tests. So, in this example, they took six numbers and found a test that it failed. Of course. Can always do that -- given 6 numbers, knowing the numbers, can always find a test that it fails. There is an old remark that some probability theory guys "didn't believe in statistics". Hmm
simonh 4y ago
A few other ways to thing about the lottery question. What are the chances this sequence would occur in the entire history of the lottery to date? What are the chances that any sequence of multiples of 9 would get drawn? What are the chances that any sequence of multiples of any number would be drawn? What are the chances that any simple mathematical progression of numbers of any kind would be drawn, again in the history of the lottery not just on one day? I play an online game called Axis & Allies 1942 Online. It uses dice for resolving battles, and in a typical game you might roll many dozens of battles and several handfuls of virtual dice for each battle. We regularly see people go on Discord or the Steam forums to complain that some extraordinarily unlikely outcome happened to them. The thing is with thousands of people playing the game, typically in several games at a time, each rolling hundreds of handfuls of dice every day, one in 10,000 odds outcomes that seem extreme are going to happen on a daily basis to someone. Often several someones a day. Every now and them one of those people is likely to go and complain about it online, so IMHO this is an expected outcome. So far this argument doesn't seem to have convinced many of the 'victims' though.
MarkusWandel 4y ago
There's a lot of math here, but what stands out for me is very simple. If you're going to play the lottery, play numbers that have the lowest odds of others playing them too. So any cute sequences, valid dates and so on are right out. You have the same chance of winning with truly random, weird numbers. But you increase your odds of not having to share.

145 points by jjgreen 4y ago

This article started off really clear and straightforward, then it dove into stuff that I clearly wasn't going to be able to follow, and upon skipping it I arrived at the explanation which was was plainly obvious from common sense.

rubatuga 4y ago
Yeah, that article really took a sudden nosedive
minihat 4y ago
>such a number is often dutifully provided to such journalists, who in turn report it as some sort of quantitative demonstration of how remarkable the event was. Apparently the conclusions are not common sense if you are a journalist.
andreareina 4y ago
Common sense can mislead, so it's instructive to make it rigorous to see whether it does indeed hold up. Not for everything or even most things, otherwise you'd be analysis paralysis sets in.
- jiggywiggy 4y ago
  In this case the author was a bit silly. He also used common sense to determine he had no way of determining the probability of a rigged event. And therefore using Bayesian probability looked fancy but didn't help learn anything new. I remember reading first paragraph and thinking wow, bayesian has a way of helping determining the odds of that, but no. Either he wanted to educate or is so involved with statistics that it became his default way of framing problems.
  - lupire 4y ago
    > Either he wanted to educate Yeah, Professor Tao has been known to try that from time to time.
Rioghasarig 4y ago
Well that's what mathematicians like to do.

ethanwillis 4y ago

And yet what Mr. Tao misses is his own assumption that the drawing of each number is truly independent.

FabHK 4y ago
They’re not independent. If the 9 is drawn, the next ball won’t be the 9.
- rippercushions 4y ago
  Mr Tao accounts for this, since (55 6) = (55/6)*(54/5)*(53/4)*(52/3)*(51/2)*(50/1)
Tainnor 4y ago
Yes, surely fields medalist Prof. Terence Tao must have fundamentally misunderstood basic probability.

fny 4y ago

Most useful takeaway from this article: If you choose to play the lotto, don't use cute numbers. Odds are high that other people will too, and you'll have to split the pot. > Part of the explanation surely lies in the unusually large number (433) of lottery winners > But on the previous draw of the same lottery ... > the unremarkable sequence of numbers {11, 26, 33, 45, 51, 55} were drawn ... > and no tickets ended up claiming the jackpot.

Semaphor 4y ago
That is actually commonly recommended for exactly that reason. Often right after recommending to not play the lottery at all ;)
- katrotz 4y ago
  Remember reading in some book on probability and statistics that one will be better off betting insignificant amounts of money rather than not playing at all.
  - Semaphor 4y ago
    Sounds wrong, that would essentially be saying one should play the cheapest level of all lotteries.
    - mellavora 4y ago
      naw, it is probably based on the assumption that the payout can be life-changing big. risking pocket change to win big is quasi-reasonable, even though it is a loosing bet. Because you don't notice the downside/drawdowns. risking more than pocket change is not reasonable, because it is a loosing bet. NB: I don't play the lottery.
      - lupire 4y ago
        Similarly, you should make small high-upside bets bets like "strike up a chat with a as stranger"

sohkamyung 4y ago

An alternative take via The Conversation [1]: > Based on anecdotal evidence from other lotteries, this number may not at all be unusual. We also need to consider the many thousands of similar lotteries drawn around the world each year, almost all of which receive no international press. While such outcomes are highly improbable for any given draw, the huge number of total lotteries means it’s actually quite likely at least one of them will produce a remarkable outcome by chance alone. [1] https://theconversation.com/433-people-win-a-lottery-jackpot-impossible-probability-and-psychology-suggest-its-more-likely-than-youd-think-191946 https://theconversation.com/433-people-win-a-lottery-jackpot...

AdamH12113 4y ago
The multiple endpoints fallacy also comes into play here. Instead of multiples of 9, it could have been multiples of another number, or any group of sequential numbers, or sequential even numbers, or prime numbers, or... The odds of a particular remarkable thing happening are very low, but the odds of some remarkable thing happening can be much higher.

tnuc 4y ago

Maybe they just got the numbers from a fortune cookie?

bananatype 4y ago

idiocrat 4y ago

That reminds me that I finally need to start participating in lotto, just to have a skin in the game. Not participating - zero chance of winning. Participating - 1 in 30 million chance of winning 4m USD.

tomcatfish 4y ago
0.000000000000001 chance - cost $0 0.000000033333333 chance - cost $4 Which one comes up on top, expected value wise?
- Epa095 4y ago
  Your evaluation of the price might not scale linearly with the money amount, making it a fault to focus to much on the expected value. Example 1: you own the mafia 1 million dollar, pay tomorrow or die. You have absolutely no way to get the money before the deadline. You have some money though, and the winner of the lottery gets 1 million. Your expected value(in $) is negative, but the chance of surviving is no longer 0 if you play the lottery. Example 2: you are a normal lower income person. You have no real chance of getting rich, or even comfortable. You can save the 4$, and maybe buy a slightly better TV every 3 years, but it will make no real difference in your life. Winning e.g 2 million on the other hand would lead to a qualitatively different (and better) life. Its not that 2 million is 2millon/4 times better than 4$, it is just incomparable. One leads to a completely different life.
  - dkbrk 4y ago
    You're not wrong in principle that you could have a utility function with positive expected value for playing the lottery, but that's misleading. Generally it is taken that money has diminishing marginal utility. This is even more salient for poor people, for whom the marginal value of $1 at their current wealth is vastly more than the value of $1 at the wealth of winning the lottery; i.e. playing the lottery is even worse, in terms of expected value, when you apply a reasonable utility function. You're correct about your mafia example, but that's highly contrived. I contest that your second example does not show what you claim; that is exactly the sort of person who should not be playing the lottery.

FabHK 4y ago

The beginning of the article is an excellent (though conventional) introduction to Bayes and Bayesian updating, but what I found original and have never seen expressed so clearly were the remarks about dishonest discussion tactics, especially the first one: Formulate a vague hypothesis H1 so broad that it is not extremely unlikely (thus has non-negligible prior, P(H1) > 0), then in the updating step sneakily introduce a much more specific hypothesis H1’ that is far less likely a priori, but that yields a high probability for the evidence P(E|H1’). If the audience doesn’t notice the bait and switch, it’ll come away with the impression that the evidence strongly supports H1/H1’ vis-à-vis H0, when it actually doesn’t. Reminiscent of the Motte and Bailey fallacy. https://en.m.wikipedia.org/wiki/Motte-and-bailey_fallacy https://en.m.wikipedia.org/wiki/Motte-and-bailey_fallacy

mistermann 4y ago
It's even more complicated than that though: "dishonest discussion tactics" suggests that the person is doing it with fully conscious awareness and intent, but this is far from a safe assumption - and, it isn't a simple binary "is or is not", and it isn't only the usual suspects who commit this technique.
- simonh 4y ago
  Maybe so, but ultimately it's up to us to take responsibility for the techniques we use in argumentation. I would agree that it's not generally a good idea to try to use an attack on the rhetorical techniques someone uses in order to win an argument. It's best to politely find a way to challenge their argument without attacking them personally for the way they put their case.
  - mistermann 4y ago
    A very reasonable take...but I wonder if there might be a substantially more optimal approach, that is possible.
pc86 4y ago
Maybe it's just my liberal arts education but discussions like this and comments like "hypothesis H1’ ... that yields a high probability for the evidence P(E|H1’)" or "evidence strongly supports H1/H1’ vis-à-vis H0" mean absolutely nothing to me without a concrete, real-world example of what H1, H1', and H0 are.

andirk 4y ago

There have been a few national American lotteries in the past handful of years with a jackpot so big ($1.5+ billion) that it was statistically advantageous to buy every number sequence; the total price of the lottery tickets was less than the jackpot. The problem with this lotto hack was that it was not humanly possible to buy that many tickets. Also, if even 1 other person won, the person would be in the hole some 1/2 a billion $ after splitting the winnings.

jmharvey 4y ago
Somewhat counterintuitively, the billion-dollar jackpot lotteries' expected values often shrink as the jackpots grow, because the media circus pulls in a lot of players and increase the likelihood of a split jackpot. It's not that uncommon for a smaller lottery to have a positive expected value. I think the logistics of buying every combination in a pick-6 lottery (on the order of 10 million combinations) would be rather unwieldy, but a pick-5 (on the order of a few hundred thousand combos) could be doable with a small team.
- lupire 4y ago
  The "broken" lotteries aren't for that kind of jackpot, for the reason you say. The broken lotteries are counting -caeds based: when a fixed priced pool is printed on the cards, and then part of the deck is exposed as losers so the rest of the deck has positive EV.
- tzs 4y ago
  Virginia lottery around the end of 1991: pick 6 from 1 to 44 giving 7 059 052 possible combinations. The challengers: International Lotto Fund (ILF), an Australian investor group with 2500 investors. ILF did't quite manage to cover all 7.1 million combinations. They only managed 5 million. They fell short because they underestimated the time it would take to buy the tickets, which they got from around 8 grocery and convenience store chains. And no, I'm not some kind of lottery history buff. This is one of those weird bits of trivia one picks up in law school. Specifically in the class I took on transnational tax around 1995, where the taxing of ILF's winnings from this hit some edge cases in US and Australian tax law that were interesting and instructive enough for the case to make the next edition of the textbook.

eyelidlessness 4y ago

The odds of N-1 numbers being “remarkable” diminishes for each number drawn. Unless remarkable can be conjured and retconned for every number combination or every Nth number drawn. Each combination is equally likely before the first number is drawn. And equally likely at the end of the draw. But odds change when you have a partial result that’s already ruled out possibilities for each drawing. You can’t have 9, 18, 27, 36, 45, 54 if you draw an 8. The odds of that are zero.

lqet 4y ago

Comment under the article: In particular, the geometry of the lottery tickets seems important. It is plausible that 55 numbers are set up in a 7 x 8 matrix pattern, with one wildcard (to produce 56 objects). Imagine that it is done as such: *, 1, 2, .. , 7 8, 9, 10, .., 15 16, 17, 18, … , 23 24, 25, 26, 27…, 31 .. then multiples of 9 are the main diagonal. That would explain frequency in an easy way.

ginko 4y ago
Looks like that's the case. See: https://primer.com.ph/tips-guides/wp-content/uploads/sites/5/2018/10/lotto-play-card-primer.jpg https://primer.com.ph/tips-guides/wp-content/uploads/sites/5...

jzer0cool 4y ago

Interesting question for this lottery draw. 1 - How about past lottery dates. Is there also a high purchase of these same sequence of numbers? 2 - Who are the 433 winners? Is there an unusual clustering of the winners? Was it mostly purchased by a single entity?

anonymous344 4y ago
This is actually the only smart question(s) to ask to find out was it a cheat
bananatype 4y ago

aaron695 4y ago

apienx 4y ago

ELI5 Bayesian theory. https://youtu.be/HZGCoVF3YvM https://youtu.be/HZGCoVF3YvM

botencat 4y ago

It’s an interesting read but the author could have just used Kolmogorov complexity to explain all this.

mvcalder 4y ago

H''''': The lottery is run by corrupt officials, who want to cover their tracks. In order to do so they want to have many associates win so no single winner is subject to scrutiny. They need to communicate to this diverse group of conspirators the correct numbers to choose in a simple way, say: multiples of 9, or anti-diagonal of ticket. Remark 4: The human mind is an amazing hypothesis generating machine. If it also knows about Bayesian statistics, it is capable of accounting for it. Paranoia is a bitch.

graycat 4y ago

For any six numbers that can exist or be drawn, once they are drawn can find something about them that is improbable. Sooooo, that found something improbable about the numbers, once they have been drawn, means essentially nothing.

kgwgk 4y ago
On the other hand, those numbers were "special" before they were drawn - which is why more people played those numbers than would be expected for "non-special" combinations.
lupire 4y ago
"Essentially" is load-bearing there. You can compute the statistics to determine what number sets are more suspicious.
- graycat 4y ago
  I don't think we are on the same page: There have to be some uncountably infinitely many statistical tests can apply to six numbers, and given any list of 6 numbers it will have to fail, be very "suspicious", according to some of the tests. So, in this example, they took six numbers and found a test that it failed. Of course. Can always do that -- given 6 numbers, knowing the numbers, can always find a test that it fails. There is an old remark that some probability theory guys "didn't believe in statistics". Hmm

simonh 4y ago

A few other ways to thing about the lottery question. What are the chances this sequence would occur in the entire history of the lottery to date? What are the chances that any sequence of multiples of 9 would get drawn? What are the chances that any sequence of multiples of any number would be drawn? What are the chances that any simple mathematical progression of numbers of any kind would be drawn, again in the history of the lottery not just on one day? I play an online game called Axis & Allies 1942 Online. It uses dice for resolving battles, and in a typical game you might roll many dozens of battles and several handfuls of virtual dice for each battle. We regularly see people go on Discord or the Steam forums to complain that some extraordinarily unlikely outcome happened to them. The thing is with thousands of people playing the game, typically in several games at a time, each rolling hundreds of handfuls of dice every day, one in 10,000 odds outcomes that seem extreme are going to happen on a daily basis to someone. Often several someones a day. Every now and them one of those people is likely to go and complain about it online, so IMHO this is an expected outcome. So far this argument doesn't seem to have convinced many of the 'victims' though.

MarkusWandel 4y ago

There's a lot of math here, but what stands out for me is very simple. If you're going to play the lottery, play numbers that have the lowest odds of others playing them too. So any cute sequences, valid dates and so on are right out. You have the same chance of winning with truly random, weird numbers. But you increase your odds of not having to share.