What Are the Odds?

[FabHK]

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

[pc86]

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.

[mistermann]

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.

[lqet]

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]

Looks like that's the case.

See: https://primer.com.ph/tips-guides/wp-content/uploads/sites/5...

[fny]

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]

That is actually commonly recommended for exactly that reason. Often right after recommending to not play the lottery at all ;)

[yeellow]

I read once about a famous mathematician who claimed that whenever he plays lottery he chooses the sequence 1,2,3,4,5,6 because nobody else would. Also it seems strange that so many people would choose a sequence that most people would call "not random" and suspicious (hence the article). Maybe those people follow the advice of this famous mathematician. Authorities should check the occurrence of this series in the previous games to see if that many people really like this sequence or if it was a one time event. Thinking about this, it would be interesting to see what numbers people choose, if there are some outliers preferred by many. I wonder if there are lotteries that publish such data?

[geysersam]

I wonder if the lottery companies takes this into consideration when setting the price of their tickets.

If all your players pick the same numbers, you as the lottery seller is in a much better spot, and you could safely lower the prices or increase the pot to draw more people into the game.

However, if you do that and people start picking random sequences you risk losing a lot of money.

[unphased]

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.

[Rioghasarig]

Well that's what mathematicians like to do.

[minihat]

>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]

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.

[rubatuga]

Yeah, that article really took a sudden nosedive

[mvcalder]

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.

[sohkamyung]

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...

[AdamH12113]

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.

[andirk]

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]

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.

[jzer0cool]

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]

This is actually the only smart question(s) to ask to find out was it a cheat

[simonh]

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.

[eyelidlessness]

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.

[MarkusWandel]

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.

[graycat]

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]

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]

"Essentially" is load-bearing there. You can compute the statistics to determine what number sets are more suspicious.

[idiocrat]

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.

[andirk]

I don't play and instead write down my selection of numbers. If I lose, I won the $3 I would have spent. If I win, well, that hasn't happened so we'll see.

[tomcatfish]

0.000000000000001 chance - cost $0 0.000000033333333 chance - cost $4

Which one comes up on top, expected value wise?

[tnuc]

Maybe they just got the numbers from a fortune cookie?

[apienx]

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

[botencat]

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

[ethanwillis]

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

[Tainnor]

Yes, surely fields medalist Prof. Terence Tao must have fundamentally misunderstood basic probability.

[FabHK]

They’re not independent. If the 9 is drawn, the next ball won’t be the 9.