The gambler's fallacy is a fallacy because of the assumed probability and the independence of the events. However, if, after flipping a coin 100 times and obtaining heads each time, I still believe the probability of obtaining tails to be 0.5, am I not making a different mistake? Is there a name for that kind of fallacy?
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25$\begingroup$ Stubbornness $\endgroup$– kjetil b halvorsen ♦Commented Apr 9, 2021 at 20:29
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14$\begingroup$ important detail missing from the question (that many answers given so far depend upon): flipping a coin or flipping a coin known to be fair ? That detail changes everything. $\endgroup$– TomCommented Apr 10, 2021 at 6:07
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4$\begingroup$ @Tom If one knows for a fact that the coin is fair (which may be somewhere between difficult and impossible), the question wouldn't make much sense. Although if you have other evidence about the fairness of the coin (like prior flips), it would be a question of comparing it to that evidence before jumping to conclusions. $\endgroup$– NotThatGuyCommented Apr 11, 2021 at 7:00
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4$\begingroup$ Insanity. By some definitions: "Doing the same thing over and over and expecting a different result." (Often misattributed to Albert Einstein, not sure where it actually comes from.) $\endgroup$– Darrel HoffmanCommented Apr 12, 2021 at 17:59
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6 Answers
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That seems to be a typical example of the Ludic Fallacy: https://en.wikipedia.org/wiki/Ludic_fallacy#Example:_Suspicious_coin
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1$\begingroup$ Thank you. Although all answers were helpful and expanded my knowledge, this one did it in for me the most unexpected direction. $\endgroup$– Igor F.Commented Jul 2, 2021 at 13:08
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N Blake's answer is great, using the language of cognitive psychology. If you're up for a Bayesian slant, you could treat this as violating Cromwell's rule.
You have a prior certain belief that the coin's probability of giving tails is $f=0.5$. In other words, you believe that $p(f=0.5) = 1$, which is the problem. This gives you no flexibility to update this as the experiment continues. Your posterior $p(f \mid \text{100 heads})$ is now proportional to
$$ \cases{ p(\mathrm{H} \mid f)^{100} \times 1 & if $f=0.5$ \\ p(\mathrm{H} \mid f)^{100} \times 0 \color{red}{= 0} & otherwise } $$
Despite this long run of heads, your posterior remains unchanged from the prior! You didn't update your mental model of the coin despite the consilience of evidence against it being a fair coin.
answered Apr 9, 2021 at 14:05
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Belief perseverance seems to fit - maintaining a belief (that it's a fair coin), despite contrary evidence (100/100 heads).
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It's called a strong prior.
The more enduring your belief in the coin's fairness is, the stronger the prior would be said to be. Might call it an absolute prior if it'd never change.
It's not necessarily a fallacy so much as an approximation, especially if the prior is strong enough that it'd be difficult to meaningfully distinguish its strength from absolution.
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1$\begingroup$ That's what Cromwell's rule is in my answer above—an overly strong prior that can never be changed because it is either 0 or 1. $\endgroup$ Commented Apr 9, 2021 at 19:44
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3$\begingroup$ It is important to call it simply a "strong prior" as this 1) is the correct answer, 2) is exactly responsive to the sense of the original question, and 3) serves to give full voice to the meaning of a prior. So many of us are mis-taught in our Bayesian methods class that the only good prior is a noninformative one. $\endgroup$ Commented Apr 10, 2021 at 4:42
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$\begingroup$ @AryaMcCarthy: Yup,
+1'd your answer as Cromwell's rule was a good thing to point out. $\endgroup$– NatCommented Apr 10, 2021 at 18:38
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This belief was examined in a series of papers on the "gambler's fallacy" and broader methods of binomial prediction under the Bayesian paradigm (see O'Neill and Puza 2005; O'Neill 2012; O'Neill 2015). These papers argue in favour of your view here – that observing more heads in a series of coin-flips should shift your belief somewhat towards having more heads in the future. Those papers referred to this (correct) belief as the "frequent outcome approach", and noted (just as you have) that the persistent belief in fairness is problematic. Those papers do not give any name to the latter error, but I agree with some of the names suggested by other commentators ("stubbornness", "belief persistence", etc.).
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I would call that a Type II error, which is anytime in science you fail to reject a null hypothesis even though it is false. If you had no other reason to think the coin is fair other than the flips you've seen, you could calculate draw a conclusion on evidence alone/ If you conclude it is fair just because you think you have insufficient evidence to conclude the proportion is different than 50/50 in the infinite limit, and if it is in fact not 50/50, that is a Type II error.
