The Gambler's Fallacy Is Not a Fallacy
kevindorst.com
kevindorst.com
I'm surprised that HN is so quiet on Sunday morning that 8 pts is enough to make #5 frontpage post for an article that fails the Wikipedia Test (”is the article better than the Wikipedia post on the same topic?”)
In the standard gambler's fallacy situation, it's assumed known that the coin is fair.
However, in real life there's always some probability that the assumption is incorrect.
One way to think about it is in terms of likelihoods, priors, and posteriors over models, in addition to the probability of an outcome conditional on a model.
So, the classical assumption is something like P(X | Mf) = 0.5 for a "fair" model Mf, and you're asking someone "what's the probability of heads?". However, there's also the possibility that the coin is actually biased, under Mb. So the actual probability of an observed sequence is something like
P(X|Mf)P(Mf) + P(X|Mb)P(Mb).
Usually we assume that P(Mf) >> P(Mb) but there must be some point at which P(Mb) becomes great enough that it would be rational to start to question that.
Implicitly there's some Bayesian estimate of P(Mb|X) that could be estimated, and some decision point where you decide P(Mb|X) > P(Mf|X).
"The 'koin' is fixed. Bet Tails."
I most certainly cannot agree with this premise. I’ve met many people who make this mistake all the time. I even have a friend who is amongst the smartest people I know who honest to god thinks he’s lucky. He believes that there is some force that allows him to either effect the next said coin toss or allows him to devine the next coin toss. It’s wild, he’s even really good at board games too, so it’d be easy to think he might be lucky too.
I have incomplete information on what a "koin" is -- I am told that it's like 50/50.
But PURELY from what I have seen, it clearly likes "tails."
Exactly why shouldn't I bet "tails?" This is pretty much the same as "yes, you can't PROVE that the sun won't rise tomorrow but that's how we act."
> Let's suppose you can be sure that one of the three particular Sticky/Switchy/Steady hypotheses in Figures 1–3 are true, but you can't be sure which.
In reality, if you don't know what's true, you also don't know that it must be one of a small set of convenient models.
Right, so in that case it isn't clear that the committing the Gambler's Fallacy is going to fare any worse than other prediction models.
This is IMHO the best kind of nonsense -- by its own "nonsensical" arguments, it shows that the textbook-derived arguments aren't really that great either. Seriously, as the article points out, a lot about probability are just tautological based on artificial assumptions -- eg. "assume a fair coin", but of course coin tosses in the physical world aren't 100% fair.
I think the author will agree with your sentiment that people assuming convenient models is the problem. If I understand the article correctly, it wasn't to make the assumption as a general statement about coin/koin tosses, but to illustrate alternative processes/models over the textbook "fair coin toss", which in itself, has been demonstrated to be not so fair in empirical studies.
Truth/Fallacy 1: There is a 50/50 probability that the next flip will be either Heads or Tails.
Truth/Fallacy 2: In the long string of flips, the probability is that there will be a 50:50 distribution of Heads and Tails, so the next flip be a value which will tend to regress the overall value to the mean of 50:50.
If you hold to Truth No. 1, you have no difficulty in believing that 1000 flips can come up 1000 Tails just as easily as the 50:50 result.
If you hold to Truth No. 2 you are guilty of subscribing to the 'Gambler's Fallacy'.
Which way do you jump? :)
There are (1000 choose 500) ways to an equal number of heads/tails and 1 way to get all tails. If you get all tails, you should probably question the correctness of your model.
Many things aren't independent. The article mentions the odds that it will rain on day depends on whether it rained the previous day. Likewise, the odds of a bus arriving in a city with a well-organized strict bus schedule is less if a bus just recently arrived.
How about blackjack? If I get an ace, are the odds still 1/52 that my next card will be an ace? Of course not.
How about the lottery? Are the odds that a scratch-off ticket is a winner different if the previous ticket on the roll was a winner? I have no idea, but I wouldn't be surprised if the lottery runners introduced such shenanigans.
I think that's the author's only point - that we shouldn't always assume independence.
But it falls apart a little because it seems to be talking about coins (even if it's actually talking about koins), which is a domain that starts from the premise that events are independent.
And that premise is empirically false as noted by the author.
Now the question is whether it's meaningful to accuse people of committing fallacies in hypothetical scenarios where perfectly fair coins are real.
The author is not making this point, and early on, he calls coin flips "Markovian" which is synonymous with "memoryless". Independence is assumed in this article.
Independence is not assumed in the "sticky" or "switchy" states in the article. The probability of the next flip depends on the previous flip.
You're right about the article.