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Agreed. As far as I can tell the author's premise is: "normally the Gambler's Fallacy refers to things like coins and dice, where you don't know the precise odds but you do know that trials are independent. But if we redefine it to refer to the opposite situation - where you somehow know the overall odds but you don't know whether trials are independent - then we find..."

> 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.

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.

> 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.

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.

Even a biased coin will flip independently, which is all you need for it to be a fallacy.

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)?

> 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?

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.

> 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?

>>How do you test a roulette wheel?

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.

No.

The house always wins, until it doesn't. And on a long enough time scale...

> How do you test a coin to determine if it’s unbiased?

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.

but the system is also at play. so the nature of two outcomes (ignoring the side case) results in a rule of play when you factor in the state changes, the ways of changing states, and the likelihood of change.

Unfortunately while this applies to (fair) coin tosses, it does not apply to the normal situations statistics is applied to.

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.

In situations where we don't know the probabilities, the 'Gambler's Fallacy' cannot apply.

Gambler's fallacy applies if we know events are independent even if we don't know probabilities.

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.

... which is the point I'm making. If you measure something, then create a policy to change the thing you measure, then measure it again. At that point measurement and policy are designed to be dependent on each other. If the measurement had given a different outcome the policy would have never seen the light of day. You can't get much more dependent than that. Stating these are independent is like saying your parents had nothing to do with your birth.

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.

> This is straight up false because you know a priori that the chance is 50-50.

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.