The gambler's fallacy is not a fallacy?
kevindorst.com
kevindorst.com
This is straight up false because you know a priori that the chance is 50-50. It’s gamblers fallacy if you know the events are independent.
If you don’t know they’re independent and you’re trying to identify the pattern that’s just an experiment.
This whole write up ignores that events are literally independent and therefore it is irrational, and gamblers fallacy, to believe that provably independent events influence each other.
[edit] > So back to gamblers. What is the gambler's fallacy? Many have suggested to me that it's the tendency to think that a heads is more likely after a string of tails, despite knowing that the tosses are statistically independent. But this can't be right––for no one commits that fallacy
Yeah they do. At the roulette table, all the time. Gamblers do this a lot, which is where it gets its name.
For example. Suppose you have bad thing X. Homelessness, soil pollution, ... whatever. You measure it. It's bad. You have policy Y. You want to know if policy Y improved X. You measure it again. The measurement is better.
Did X improve? Correct answer is: you don't know!
The problem: suppose policy Y is "fake measurement X". Then it's obvious why X didn't improve. If you think about it, you'll quickly realise: X and Y need to be independent. But Y was a response to measuring X! They cannot ever be independent! The very existence of policy Y is dependent on X!
So the standard way statistics is used, to measure something, then make changes to improve it ... is a fallacy. Exactly because it causes the gambler's fallacy to not apply anymore! The gambler's fallacy only applies to situations where successive measurements are independent. And if you actually use your statistical measurement, well that's an excellent way to make successive measurements dependent, and makes measurements invalid.
Of course, when we don't know the baseline probabilities, its possible that updating one’s prior in the opposite direction of the experienced results ends up accidentally getting the right estimate are some point before heading off in the wrong direction, but that's a “right answer by wrong logic” coincidence, not Gambler's Fallacy failing to be wrong.
And if you will allow me a different statistical fallacy: this is 99.9% of what we do with statistics. You can measure once, then change things. Then you can never measure the same thing again. Stating "we saw 5% unemployment, then did policy X, now we see 4% unemployment. Success!" ... that's mathematically wrong and a statistical fallacy.
The issue is: this is how government is run.
Odd, it's almost like you stopped reading before you got to the meat of the argument, which is that correlated events can still have a 50% chance of occurring on average.
We very carefully weren't told in the article that koin events are independent, this is a setup for the entire point of the article.
Not necessarily. As the article notes, the way humans normally toss coins, the coin has about a 51% chance to land on the side that was facing up at the start of the toss.
The more general point here is that the odds for a coin toss are not a property of the coin by itself; they are a property of the coin and the process that is used to toss it. IIRC E. T. Jaynes in his book on probability theory described experiments with a robotic coin flipper where it was found that the odds could be skewed quite significantly by properly designing and calibrating the flipper.
> If you don’t know they’re independent and you’re trying to identify the pattern that’s just an experiment.
Yes, I think most of the article would be better described as how to experimentally test for the actual odds of a process like repeated coin flips, given several possible hypotheses about how the process works.
> Yeah they do. At the roulette table, all the time. Gamblers do this a lot, which is where it gets its name.
Yes, the biggest missing piece I see in the article is any actual discussion of how actual gamblers in the real world behave.
Oh. You know a priori.
How?
How do you test a coin to determine if it’s unbiased?
How do you test a roulette wheel?
If we took a coin that you ‘know’ a priori is unbiased, and we flipped it a million time, and it came up 98% heads, would you still stubbornly claim the next flip is just as likely heads as tails? I doubt it… at some point (probably a lot sooner than a million throws) you would begin to perceive a possibility that your a priori belief was wrong. That’s the issue… is it rational, at some point, to doubt your a prior belief? If the answer is yes for a run of 183,829 heads, then it must also be yes — although with much less confidence — for a run if 8 heads. Or 5 hesds. Or even 2.
If you flip a coin a thousand times that you aren't sure is biased and it comes out heads each time, you might reasonably decide it is probably a two-headed coin and update your expectation to a ~100% chance of heads.
But that's the opposite of the gambler's fallacy, which would think that after a thousand heads it's practically guaranteed to come up tails the next time.
So which is it- is the string of heads evidence of bias (you should expect more heads) or is the gambler's fallacy true (you should expect more tails)?
It's almost impossible to bias a coin. To bias a coin, you have to both design it carefully and flip it in a specific way that's incredibly obvious.
> How do you test a roulette wheel?
There are statistical techniques that give you a confidence that the roulette wheel is unbiased. You keep spinning it again and again and plugging the results into the equations. They spit out numbers that trend towards (or away) from 100% certain. At some point, you accept that it's close enough to 100 (or 0) and move on with life.
It's actually very simple - you place it in a public place, allow people to bet on it in the usual fashion (with the usual odds), and wait. If whoever is running the wheel ends up bankrupt, it was a biased wheel.
The house always wins, until it doesn't. And on a long enough time scale...
You know what I really wouldn't do? Conclude that it was due for heads, with greater than 50% probability. Gambler’s fallacy isn’t concluding bias from a pattern, it's specifically believing their is an interdepents between events such that disfavored results from a prior series have a higher probability of occurring next.
You are talking about a completely unrelated process of updating priors about bias of independent events.
I hear you. I understand the point you are making.
But, the two issues are not entirely unrelated…
You are assuming - a priori - that the tosses are independent. That assumption can be tested. Right? How would you test it? Wouldn’t your belief in the independence of throws be affected by a long run of heads? What if there was some mechanism whereby a previous head slightly increase the chance of it being heads again? Even if we couldn’t conceive of what the mechanism was, certainly we could test for it, right? How would we do that?
The bit where they present their main evidence against the gambler's fallacy:
> So back to gamblers. What is the gambler's fallacy? Many have suggested to me that it's the tendency to think that a heads is more likely after a string of tails, despite knowing that the tosses are statistically independent. But this can't be right––for no one commits that fallacy. After all, knowing that the tosses are independent is just knowing that a heads is not more (or less) likely after a string of tails
I think "no one commits that fallacy" is plainly untrue. There are various games (Gacha games, https://en.wikipedia.org/wiki/Gacha_game) where the outcome is random independent probability. In fact, due to gambling laws, these games have to publish exact probabilities.
Despite there being clear evidence, people will commit a combination of the sunk cost fallacy and gambler's fallacy and convince themselves to spend more money on more independent random trials because surely they're almost there.
We've seen similar things with lootboxes with published odds.
We see similar things with casinos that contain various games of true independent random probability.
They key difference here is between people rationally knowing that those odds are independent, and them being able to bridge that to their irrational belief that they're special, or lucky, or that they're "owed" luck. People are not rational, people can hold conflicting things in their head, and the gambler's fallacy is one of the common irrationalities a lot of people act on.
Interestingly, the Wikipedia page you linked says that in some games, the player's chance does go up with each roll.
(Ctrl+F for 'pity')
> They key difference here is between people rationally knowing that those odds are independent, and them being able to bridge that to their irrational belief that they're special, or lucky, or that they're "owed" luck.
I think in part it's just greed as well, they think they can have that money that all those unlucky people have lost. This greed then gets rationalized as luck/skill/smarts in a way that's not dissimilar to how people rationalize joining pyramid schemes.
These pre-determined fixed odds make it such that the games support the opposite of your argument.
If outcomes were independent probability then it would be possible to for example win 100 games in a row (with astronomical probability, but it's still possible). In fact the machine owner neuters it so that the house won't lose in such a streak.
Similarly, the machines have a fixed payout %. If each outcome were truly independent, it would be possible for the machine to never pay out, just like it's possible to flip Heads any arbitrary number of times in a row.
Anyone sucked into these games thinking 'the next one will hit' is not committing to gambler's fallacy because these outcomes are not independent.
Slot machines are based on RNGs.
> If each outcome were truly independent, it would be possible for the machine to never pay out
It is possible for a machine to never pay out. That would probably trigger an audit to see if there's a bug, but there is no guarantee that th
So if £1000 is paid in during the day, the machine by law must pay out at least £700 in prizes. This is not allowed to be fully random, the odds of each game are not independent because they must by law ensure the minimum payout.
It's semi-common knowledge that the way to win at these machines is to watch them until you have seen someone else have a long losing streak, then take your turn when they leave as the machine is now 'behind' and overdue a payout.
Nevada has the opposite rule: machines must only incorporate chance when paying out and the theoretical payout is what the gaming commission analyzes. They can't dynamically adjust odds to favor a preferred payout. [1]
[0] https://www.gamblingcommission.gov.uk/authorities/guide/page...
[1] https://gaming.nv.gov/modules/showdocument.aspx?documentid=4...
> If outcomes were independent probability then it would be possible to for example win 100 games in a row (with astronomical probability, but it's still possible). In fact the machine owner neuters it so that the house won't lose in such a streak.
The possibilities you say cannot happen do happen. Let's look for example, at "Another Eden" https://en.wikipedia.org/wiki/Another_Eden
I don't really know anything about it, but it was one of the first on wikipedia's list of gacha games that I googled "pity system" and got "no it doesn't have one".
Someone has made a simulator for it. If I go to https://ae-encounter-sim.github.io/ and click on "Weapon Discovery Manifestation", it shows me two characters on "rate up" (more common). The official probability of getting each of them is 0.8% chance for each "encounter".
I'll click until I get the first character (5star Melina apparently) and post my results:
1. First time, 360 encounters (36 10 bundles), cost of $959.88 USD.
2. Another try, 480 encounters, cost of $1,200 USD
3. Last try, 130 encounters, cost of $319.
It is very possible to get the desired outcome in 1 "encounter", or to run out of money first and never get it.
> the machines have a fixed payout % [that isn't independently random per trial]
There's no fixed payout if there's no pity system. Why would there be, they're just giving away jpeg images.
A lot of casino machines also don't necessarily have fixed payout. Over long enough time periods, the casino still wins, they don't really need to worry about incredibly improbable events. For those, they have insurance anyway.
> not committing to gambler's fallacy because these outcomes are not independent.
The simulator is supposedly accurate, and I expect the person who researched the probabilities to build it is likely to know what they're doing. Looks pretty independent to me: https://github.com/ae-encounter-sim/ae-encounter-sim.github....
I mean, in the introduction the author gives a standard definition of the Gambler's Fallacy, but then he immediately says "but it can't really mean that, because that's clearly irrational". It's not very hard to prove something is rational if you start by rejecting definitions that are irrational!
Knowing an outcome would always be 50/50 a gambler would be inclined to make the 'fallacious' assumption that string of heads would likely to be yield heads.
The argument made is that someone would not make that assumption! Because it's irrational.
And then through a very long winded and unnecessary example ... posits another kind of coin toss ('koins'), where the 50/50 ratio is not necessarily known, it might make sense to actually bet heads after a string of heads.
The 'fallacy' in the thesis, is that even when gamblers know the odds which is to say, rationally, we know it's 50/50 ... we will still, intuitively want to be to on heads, after a string of heads.
Our 'instincts' want to see dependent results where there are none, which is the basis of the very real fallacy.
I don't know why pages and pages were focused on creating a contrived example when a one sentence description might have sufficed, maybe I missed something ...
I think you could similarly dismiss any formal fallacy. The fact that X implies Y and some accept X (coin tosses are independent) but not Y (results are not more likely when "overdue") is what makes it a fallacy.
From then on the post redefines gambler's fallacy to a scenario in which you know the outcome percentage but not if results are independent. Still an interesting post, but not really what I've seen meant by gambler's fallacy.
I do think there's an interesting question that's raised by the essay though, which is at what point does a string become so inconsistent with a certain probability distribution that you have to question whether things are as they are assumed? That is, let's say you go into a casino and you keep getting wins. At what point do you say "this is so improbable under X probability of wins that something is off"? Gaming regulators must have some definition of this.
And yes, I realize that after n flips of Heads, the probability of the n + 1th flip being Heads or Tails is equal. i.e. P(Heads) ^ n * P(Heads) = P(Heads) ^ n * P(Tails), because P(Heads) = P(Tails) = 0.5
It doesn't make sense because the events are independent. It's not a better or worse bet to pick one or the other. It doesn't make sense to think one is more than 50% likely because of a series of other independent events when you know the odds a priori.
How do you know how many Heads and Tails the coin has flipped in its lifetime? Maybe its previous owner flipped a bunch of Heads. Maybe it flipped some Heads while it was jangling in your pocket. And if you don’t know, how the hell do you expect the coin to know?!
The catch is: only for an infinite number of throws and monotonicity is not guaranteed.
If you knew, a priori, that 75% of the next 4 flips will land on H, and you've seen THH, you are justified in guessing H. In this scenario, H was actually due.
I think it's clear that real coins make no such promises.
If that doesn't help your intuition, maybe try this technically incorrect view. Those 'extra' Heads you feel are due can appear at any point in the entirety of the future. What are the chances they show up in the next five minutes, never mind the next billion years? You're looking for a handful events spread over the entire rest of time itself, after all.
As you mentioned, you don't see the convergence to 50% with any finite sample size, because that behavior is only guaranteed at infinity. All this to say, infinities are weird. 10,000,000 + infinity exactly equals infinity.
This is wrong. It takes considerably fewer flips than "infinity" to see distinct convergence here.
If the coin is truly even - you have a binomial distribution, and the probability that you are far out to either edge of that distribution (nearly all heads, or nearly all tails) is vanishingly small with just 1000 flips.
Without any such guarantee of the larger result, all you have is a 50/50 chance for the next flip. With infinity, you can pick out a portion of the sequence with an unusual number of Tails and expect there to be an offsetting number of Heads elsewhere in the full (infinite) sequence.
This is incorrect. The convergence is to: (current heads + 0.5(n flips)/
(current flips + n flips).
For a gambler flipping coins, If they are neutral at the time of prediction, you expect them to end neutral.
If they are up flips when you enter the room, you expect them to the run up flips.
A random number <=0.5 is heads, >0.5 is tails.
Flip 100 million coins.
After you get 10 of the same result in a row, count how many times the 11th is the same as the first 10.
The odds of getting 12 Heads in a row is quite low.
The odds of getting 12 Heads in a row when you’ve already observed 11 coin flips land heads up is 50%.
Knowing that the odds of 12 Heads in a row is low, that you’ve seen 11 Heads in a row, and that - if you observe the next flip land heads up - you’ll have observed something quite rare, doesn’t influence the next coin flip.
The odds of getting the sequence HTTHT are exactly the same as the odds of getting HHHHH, and similarly the odds of getting HTTHTH are exactly the same as HHHHHH.
Another way of thinking of it is that there's roughly a one in a thousand chance of getting 10 heads in a row; there's a one in two-thousand chance of getting 11 heads in a row, but if you've already gotten 10 heads in a row, you've already ruled out the ~1998/2000[1] possibilities that can't ever result in 11 heads (because they didn't start out with 10 heads in a row), so you only need to rule out one of the ~2/2000 remaining possibilities.
1: Actual numbers is 2046/2048 and 2/2048
@dang: IS HN meant to change the timestamps of old comments when an article is moved back onto the front page?
But what about the chance that the "koin" is actually controlled by someone malicious? The liklihood of getting heads would have nothing to do with past events; it would just depend on how much money I bet and on which outcome. A paranoid gambler might easily assume this option is on the table.
Wouldn't that break the link between "<50%, =50%, >50%" and the concepts of "Steady-ness, Switchy-ness, Sticky-ness"? If so, can you conclude anything about the behavior of the koin since the probabilities of "Steady-ness, Switchy-ness, Sticky-ness" (as concepts) no longer add up to 1?
> T H T T T T T
> How likely do you think it is to land heads on the next toss? You might look at that sequence and be tempted to think a heads is "due", i.e. that it's more than 50% likely to land heads on the next toss.
> That's irrational. Right?
> Wrong. Given your evidence about koins, you should be more than 50% confident that the next toss will land heads; thinking otherwise would be a mistake.
Given the evidence I have about koins (the sequence) I would actually make the assumption that it's most likely for the next toss to be T. And that koins is not a fair coin.
In a more gambler relevant context, if a roulette ball ends on red 10 times in a row. It's more likely to end on red on the 11th time as well, the implication being that the roulette is not fair.
I would argue that expecting people not to do this even though you see it every other day on HN is a variant of the Gambler's Fallacy :p
No, Symetery is what drives the long term 50% rate, not switchyness or stickyness. observing a 50% rate in a small finite dataset is evidence for switchyness, but the size of the data set is not included in the analysis provided. Alternatively, the variability of a dataset can be evidence for switchyness.
Also, say you're about to flip a coin 100 times. Then, your future-self advises you that you're 50% likely to get Heads on any particular flip. Then, sure, that'd seem to imply that you're going to get 50 Head's and 50 Tail's, so switching would often make sense as getting one result would tend to reduce that result's likelihood. In particular, you should be able to guess the last flip with certainty, as you know that you ought to get a total of 50 Head's and 50 Tail's.
However, if you're flipping a coin 100 times and estimate a 50% likelihood of Head's, then that's a different scenario. I mean, yes -- you're still working under the assumption that Heads is 50% likely, but it's not the same thing. You wouldn't know that the final distribution would be 50 Head's and 50 Tail's, such that getting one result wouldn't have the same implications.
Finally, "the gambler's fallacy" is named after a stereotypical scenario wherein a gambler who keeps losing argues that they'd be a fool to quit because they're "overdue" for a win. In that case, it does tend to be a fallacy. So if you come up with a situation where such an argument wouldn't be a fallacy, then presumably you're looking at something other than the gambler's-fallacy.
https://news.ycombinator.com/pool
Selecting “past” at the top of this submission can confirm this. It shows this submission with the same submitter, score, and number of comments but dated 4 days ago.
Dang wrote about it at that link.
The first is the average of results. So 1 for heads, -1 for tails. Plot the running average over flips. It will start out all over the place. Eventually you will end up with a nice steady line near zero.
Next plot the running sum of the flips. The line will wander all over the place. At any particular time it might be a long way away from 0. In some cases it will never return.
So the average operation creates the tendency towards zero. The result can be useful sometimes, but it in no way helps predict the coin flips.
I can challenge any of you into gambling, using some JS code we agree that is not buggy, using random.org's API, here is my rules:
- we toss a coins in a row
- we do not gamble until a string of tosses is the same side
- the toss after I apply the gamble's fallacy (I'm stupid LOL), I choose the opposite side, we gamble who ever wins take a dollar
- we keep doing it until one of us go bust
Anyone up to the challenge?
If you really believe in the gambler's fallacy, then there must be some greater-than 50-50 chance of the opposite side coming up. Offer me some odds (say 5/4) and define some table-stakes (thus removing the advantage of having a higher bankroll) and I'll definitely take you up on it.
[edit]
The random.org API seems unnecessarily complex, non-reproducible, and requires trusting whomever runs the JS. I propose an alternative of us each picking a value equal to the key+nonce length of ChaCha20 and pre-sharing the sha512 hash of that value. We then reveal our picked values, XOR them together and use ChaCha20 to generate bits for heads/tails.
In essence I will be gambling that there will be more 20 in a row same side than 21 in a row same side, which might be probably true, I have to test it.
Edit: in reply to your edit, I will do this "virtual" bet and update this comment with the result. I will try to tweak it to be in my favor, if there is no way to do so, then I guess the gamble fallacy is a real fallacy.
PS: not an actual bet, just tinkering stuff (betting is haram).
After the 20 have landed the next flip will still be 50/50.
Just as 20 perfectly alternating heads/tails/heads/tails will lead to another 50/50 chance, or any other potential pattern the past may have held.
I think you're missing a subtlety here.
It's true that 20 in a row is twice as common as 21 in a row.
But 20 in a row is exactly as common as 21 or more in a row.
When you bet on that coin toss, you're not betting on 20 vs. 21. You're betting on 20 vs. 21+. So it's 50:50.
For every thousand times you reach 20, the bet that it ends at 20 wins about 500 times, and the bet that it goes above 20 wins about 250 + 125 + 62 + 31 + 16 + 8 + ... = 500 times.