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Comment by lurquer | Hacker News Reader

> 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.

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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.

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