Consider two experiments concerning similar fair coins(*):

  1. Throw the same coin N times and observe the outcome.
  2. Throw N similar but different coins 1 time each and observe the outcome.

(*) One can equally say "similar fair throws of coins" instead of "similar fair coins"

Experiment is used in a loose sense of doing something that has some outcomes.

N can be as large as we want it.

Similar means that the probabilities of heads have same value, ie p_1=p_2=..=p_N=p. Fair means p=1/2.

As far as we know there are physical models of the above processes.

According to mathematical literature the two experiments are identical. In fact the first experiment is always reduced to the second, which is the one used to model the event space.

But is this accurate?

For example, let's say we want to estimate if fairness assumption is valid:

  1. In the first experiment, throwing the same coin N times and getting an outcome of all heads is increasingly problematic for the same fair coin.
  2. Whereas for the second experiment, getting an outcome of all heads, is not problematic for any coin, simply thrown once, thus not problematic for the whole outcome.
  3. If we say that the first experiment is the same in this sense, that N throws is simply a series of one throw, then either throwing N times looses meaning, or by same token as point 2) we have no reason to infer the coin is not fair by observing any long sequence of heads, which seems paradoxical.
  4. If the sequence is all that matters and not the coins themselves, then we can infer in second experiment that all coins are not fair, simply by one throw of each, which seems paradoxical.
  5. if the sequence is all that matters and different coins can really be anywhere at anytime simply thrown once, this raises questions how the different coins are synchronized so that the sequence satisfies certain statistical properties which are similar to those of one and same coin which seems paradoxical, whereas the same questions are not necessarily raised for one and the same coin.

(**) What point 1) means is elucidated in point 3) which is its dual. It means statistically inferring fairness or not based on long sequence of data. Similarly for other points.

What is your opinion? Please elaborate on it.

Possibly related:

  1. Ergodic Hypothesis
  • 1
    Concerning the update. Well the p-value obviously has implications on the shape and makeup of the coin, like if you want it to be fair coin then you'd kinda expect at least some sort of symmetric shape so as to not favor one side over the other. So if you would run this experiment in the real world that would absolutely matter. But in terms of the thought experiment all that matters is that p-value, because that is call that actually causes the result.
    – haxor789
    Commented Jul 19, 2022 at 13:15
  • 1
    We can agree that what is similar in the coins (eg weight, shape, symmetry, etc) is summarized in having same value of their p properties
    – Nikos M.
    Commented Jul 19, 2022 at 13:35
  • 2
    Yes in the end all that matters about the coin is summarized in that property of the p-value. Assuming the distribution of digits in pi is normal you could also idk start at the 1000th place and ask whether the number is <5 or >5 and you'd have something that has the same property of a coin flip. So it's not really paradoxial, but interesting how these things that are not at all connected (or at least are not obviously connected) share a common property.
    – haxor789
    Commented Jul 19, 2022 at 13:48
  • 2
    Your point 2 (not problematic for any coin implies not problematic for the whole outcome) is not valid. Individually it's not problematic, but if you collect, through a random process, the outcomes of N different coins and you get N heads, that's problematic. As the probability of collecting N heads is quite small when N is getting large. The process of collecting the outcomes is what ties these things together. For example, if at specific time you collect the outcome of N fair coins all over the world and discover N heads, you may suspect there is some global influence that affect the outcome
    – justhalf
    Commented Jul 20, 2022 at 11:08
  • 2
    I posted an apology after I rolled it back. The deletion of those posts, obviously for the benefit of the community, were meant to witness that I have no strong feelings about this post in anyway. Given the ruffled feathers and the strong characterization of my actions as vandalism, Ill be sure to remain aloof of the OP in the future. I have no hard feelings, and I hope this young man feels his rights have been protected. : ) There was no ill intention, Herr Meister.
    – J D
    Commented Jul 23, 2022 at 0:50

10 Answers 10


Mathematics kinda "solves" this identity problem right from the start by an axiomatic definition. In that it usually states that fair coins are indistinguishable from each other. So any fair coin behaves like any other fair coin.

So if you had an imaginary bag of coins and you'd grab either the same coin each time or a different but in any way shape or form identical copy of that coin, you should get the same result. If you don't, then you're premise is broken because at least one of the coins is unique. Obviously only applicable in the limit of infinite or at least very many coin flips as each individual flip is random.

The conceptual problem is that in this thought experiment you're not really running the experiment. You don't throw the coins and you don't get any real result, what mathematics is doing here is just constructing a set of N events with a certain p-value. And as the elements of the set are defined to be indistinguishable, it doesn't matter if it's the same event N times or N times an event that is indistinguishable from the rest or whether it's a mixture of the two or whether you mixed up the order from a previous such experiment. As the coins/random events are defined to be indistinguishable that doesn't change the experiment. Obviously that only really works in the real of pure math as any attempt to physically replicate that will face certain struggles, like damage to a coin over time, coins being just slightly different from each other, different initial conditions, idk training in terms of how to throw a coin or how to catch it to determine the result and whatnot.

Also obviously that identical nature collapses if you want to prove the fairness of one particular coin, but then you're starting from a different question/assumption. In the first case you take 2 things for granted:

  1. All fair coins behave indistinguishable from each other
  2. Any fair coin's probability to show heads/tails is 50%.

Whereas if you want to test a coin, you're not actually flipping a fair coin, but you're just flipping a coin and want to test from it's properties whether it's a fair one.

Also no getting all heads on throws with multiple coins is (given the axiom of indistinguishable coins) as unlikely as getting it for one. Also that is not to say that it is impossible and for small N it is often more likely than people expect it to be. But obviously in the case of just 1 coin, it would tell you something about the probability of fairness of that one coin, whereas in the case of multiple coins it would tell you something about the probability of fairness of the whole experiment not necessarily any coin in particular.

So in both cases it would tell you something about the fairness of the experiment and if that is all that matters to you, then you could argue that under those constraints, they would be identical.

TL;DR: So essentially in math they are identical because math defines them to be identical. Also the concept of time has no significance in that example, it's all about the set of p-values. The only way to mathematically model N times is through a set of N copies.

  • Ok I got it what you mean by axiom, you mean the question. But isn't this at least one basic difference between the two experiments?
    – Nikos M.
    Commented Jul 19, 2022 at 10:01
  • On the other hand if we assume identity doesn't this postulate an "invisible hand" that choreographes different coins, so they act as a single coin?
    – Nikos M.
    Commented Jul 19, 2022 at 10:05
  • 1
    Edited the answer. Yes this would make the two experiments different but you're also asking a different question now and making different assumption (give up certain assumptions). And in that case the identical nature would collapse (in math aswell). And it's not that you'd postulate it's "the same" coin, but that you can't distinguish between coins so it might as well be "the same" coin (for all intents and purposes that matter to you). I mean you could use very differently shaped coins as long as the relevant property is preserved.
    – haxor789
    Commented Jul 19, 2022 at 10:09
  • It is not necessary to assume fairness in one case and not the other. Coins may still be taken as similar without knowing their probabilities, only that they are the same. So what you phrased s irrelevant.
    – Nikos M.
    Commented Jul 19, 2022 at 10:15
  • 1
    To make the claim in math that these two experiments are indeed identical, you are making the claim that both experiments are handling fair coins. ONLY under this premise does this claim make any sense. While testing the fairness of one coin in particular, obviously doesn't make that claim, because that's very much what is in question. But that also means that the experiments are no longer identical because the multi coin example is no longer suitable for the task at all.
    – haxor789
    Commented Jul 19, 2022 at 10:18

Consider the difference between heads and tails, typically the heads side will have slightly more weight in the middle, because of the imprint of a head there. But then the question is, how many coin flips would be required to detect that, and prove a difference that is above the confidence interval? Physical constraints always apply in practice, because the timespan of the universe is finite. No choreography required, only indistinguishability in practice, which we call fairness (note, in physics the physical impossibility of distinguishing like particles or not, has consequences for the behaviour of bosons vs fermions, we work from first principles not observation to determine probabilities).

Real coins have damage and imperfections, that will make them slightly differently biased towards one side or the other. You might expect these to impact both sides equally, so N different coins will be fairer than N times one coin, unless there is a local cultural factor like rubbing one side for luck, or playing games with particular sides facing up. The difference in weight due to printing on the coins, could introduce a systematic error, which enough flips could detect and correct for. One particular coin could also be tested for fairness, but, it might experience damage or wear in the process.

Using N brand new coins, and parsing the data for bias from the print on them, would be the most fair option.

Taking the limit of N to infinity as logical not physical:

Is it possible to flip a coin an infinite number of times and never land on tails?

The 'lie' of counterfactuals as the basis of probability theory:

If my parents hadn't gotten married, who would I be?

  • 1
    I appreciate the analysis of how the two experiments will be practically different, which is of course correct. But I ask something in the sense of "a priori" difference.
    – Nikos M.
    Commented Jul 19, 2022 at 12:20
  • See updated question
    – Nikos M.
    Commented Jul 19, 2022 at 12:46
  • 1
    @NikosM.: 'A priori', if you define them as indistinguishable they are indistinguishable. That should be obvious. It is a category of definitions, not of observable reality. Testing a coin for fairness, implies observations.
    – CriglCragl
    Commented Jul 19, 2022 at 12:46
  • 2
    This answer is correct, but could use a summary. "You are testing two different propositions: 'Are coins of this general configuration fair' vs. 'Is this coin fair'. Empirically, the result in both cases implies a strong 'no'. You are dismissing the result in both cases based on a prior belief in the fairness of coins. But real coins in our world CAN be unfair, although pretty much never to the degree shown in your thought experiment."
    – Dcleve
    Commented Jul 19, 2022 at 14:27
  • Please see updated question, you have all the time to update your answer too.
    – Nikos M.
    Commented Jul 20, 2022 at 10:20

A key fact in this hypothetical experiment is the idea that flipping one coin right next to another coin being flipped will not affect the outcome. This allows the case where n coins are flipped once to be treated as n independent events.

Note also that (equivalently) a repeatedly-flipped single coin cannot be affected by the outcome of any previous flip, which also allows the case of one coin flipped n times to be treated as n independent events.

And that means that the outcome of n coins flipped once and one coin flipped n times are statistically indistinguishable.

  • 1
    Of course but this does not address the objection of estimating p from a long sequence and finding discrepancy with fairness. It is problematic for one while not for the other
    – Nikos M.
    Commented Jul 19, 2022 at 16:51
  • I agree with this answer. Commented Jul 20, 2022 at 0:27
  • Please see updated question, you have all the time to update your answer too.
    – Nikos M.
    Commented Jul 20, 2022 at 10:20
  • Remember that statistical independence does not necessarily mean causal independence.
    – Nikos M.
    Commented Jul 21, 2022 at 20:28

"Testing an assumption" is not part of mathematics. In mathematics, if you assume something, then for the remainder of the argument its truth is absolute. So if you want to model "testing a coin to see if it's fair", then you should start with an appropriate mathematical model. Here is one:

First, estimate the prior probability that the coin is fair, before you start doing any experiments. This might be something like 99.99999% if you pick a coin randomly from your pocket, or 80% if you are given a coin by someone whom you suspect might be tricking you.

If you flip the coin a bunch of times and get all heads, then each successive heads is evidence that the coin might be biased, and you should update your prior probability accordingly, and Bayes' theorem gives you the right formula.

Are your two experiments mathematically identical? If you know that either all of the coins are biased or none of the coins are biased, then yes. If it's possible for only some of the coins to be biased, then no.

  • I upvoted this answer, as a possible argument. Instead i would prefer to say that the two are indeed equivalent only in the limit as N goes to infinity. Then this argument might make more sense.
    – Nikos M.
    Commented Jul 21, 2022 at 14:03


2022-07-21 Pursuant to back and forth with the OP, the question has substantially been refined and this answer no longer directly applies.

Short Answer

From the perspective of the philosophy of science, your experiment is rather poorly designed, and so it's difficult to provide a meaningful response. Let's provide some shallow philosophical analysis.

Long Answer

Inadequate Problem Specification

Are these random experiments the same?

Well, in philosophy, the first question is, what is "the same"? Clearly, from a metaphysical perspective, one might start with ideas revolving around the Identity of Indiscernibles. If you arrive at simple answer of yes or no, then you're not really doing philosophical analysis, at least until you have a well articulated question on exactly what you're after.

For instance, the distinction in answers has been drawn between an mathematical model that models actual physical coins (no, since coins are subtly different physically), to declaring the coins fair. Here, you are manipulating variables related to mathematical definitions in your experiment. You declare they're the same in the mathematical literature, but is that a presumption that the goal is to show that the events are independent? If that's the case, are you making an argument for an interpretation of probability that supports frequentism? Or is there something else at play? For instance, to a computer scientist, they are certainly not the same, since one scenario is serial, and the other is parallel, and parallel and serial are certainly an important distinction in communication and computation. In a computational simulation, for instance, even with a fair coin, these experiments would not be the same.

Analysis of Experimentation

But, also, what exactly is it that you mean by experiment? Are you talking about a computational simulation? Do you intend to use a physical apparatus? Or are you firmly in the territory of Gedankspiel? And if so, what sort of distinctions do you make regarding each? And since experiments are generally used to support or refute a claim, what is your hypothesis to begin with? Surely you recognize that 'Are these the same?' is almost a meaningless question without the explication above.

But is this accurate, at least philosophically? What is your opinion? Please elaborate on it.

One philosophical opinion might be, you are not drawing almost any substantive distinctions in your problem specification. You're using the word 'experiment' much like the man on the street uses the word 'theory'. Philosophers of science since the time of the logical positivists have greatly increased the sophistication of just what experimentation means. From WP:

An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried.

Where is your hypothesis? Where is your probabilistic statement? What are your standards of evaluating efficacy? Where are you delineating how this is novel, or through what means are you observing and qualifying the results? You simply don't address any of this in your 'experiment', which means you have all sorts of assumptions. Until you state your assumptions, you are pushing ambiguity, and you will continue to have ah-ha moments as the answers that come to your question help you articulate your own assumptions.

Did you notice that in your edits, you go from acknowledging the fair coin, but then follow up with a second edit about an ergodic hypothesis. The first is an abstraction, and the second has to do with generally non-coin physical processes. So we're back to is this an idealized mathematical experiment or a physical experiment? From a philosophical perspective, what is your model of explanation? Do you buy the DN model? Are you in the camp of the hypothetical-deductive? Are you aiming to confirm, verify, or falsify some proposition? None of this stated in your experiment.

Metaphysical Presupposition

We could even take the analysis to the next level by examining your metaphysical presuppositions. Are you a scientific realist? Do you accept Quine's views on a naturalized epistemology? Do you draw a distinction between psychology and philosophy and if so how? What is your understanding of theory-ladenness? You haven't even scratched the surface of the metaphysics that revolves around the coin flip from a philosophical position. It's arguable you could simply attempt to rest the context of your efforts in some generalized notion of physicalism. But you'll find that creative thinkers can challenge aspects of physicalism philosophically.


You asked for a philosophical analysis regarding your question, but do you really want it? Is your goal to understand the philosophy of experimentation and science? To come to some meaningful answer regarding a specific hypothesis? Are you just spitballing to better understand your own psychology of science? "Are these random experiments the same?" might be seen as an essentially contested question, unless you drill down into the finer vocabulary of probability, mathematics, and experimentation, your question is exporatory and probative, but not significant in the context of the philosophy of science for real-world experimentation suitable for publication. And that's not meant to be an insult, but to illustrate that the philosophy of science is a rather complex subject spanning from the thoughts of the Ancient Greeks through the mathematization of science right through logical positivists to the post-Kuhnian practice and theory.

  • 1
    @JD -- I don't agree with your summary. The experiments are pretty good informal exploratory excursions to see if fairness applied to coins generally, or to one specific coin. AFTER doing either experiment, one would then want to do the other, then a variety of additional experiments, to see what the limits are of this apparent breach of fairness. Then repeat both experiments, several times, to see replicability, with controls that appear to fall outside the bounds of the breach of fairness.
    – Dcleve
    Commented Jul 19, 2022 at 18:08
  • 1
    If there ARE no such bounds among coins, then one should then start to look at dice and other randomizer methods, to see if probability is varying locally, and try exploring physics, to see if things like gravity, or elasticity of impact, are behaving anomalously. Science is initially an exploratory process, which gradually transforms into a formulaic bookkeeping process as one better characterizes a phenomenon, and want to nail down an effect to 10 significant digits.
    – Dcleve
    Commented Jul 19, 2022 at 18:11
  • 1
    @Dcleve You're free to disagree. When I taught high school, this was the level of discourse appropriate for freshman mathematics. It's not that there isn't that there isn't the beginning of a sophisticated experiment here. It's that the language muddles through philosophical distinctions. If this a physics question about stochastic processes and probablity, then it should be asked in PhysicsSE. If it's a question about proving frequentist notions of independence of events, then Math. (Where it was closed already). It asks for the philosophical analysis of the experiment. I provided it.
    – J D
    Commented Jul 20, 2022 at 4:28
  • 1
    Of course the question has merit. You started by intuiting a fascination aspect of the universe, you translated into reasonable lay conversation, and then you refined to push it ahead. That's what all thinkers do. You have 2 votes for closures. I didn't vote for closure; I asked you to reflect on a better version. I'll reread and answer again in a different response.
    – J D
    Commented Jul 21, 2022 at 22:04
  • 1
    @JD for what is worth, this answer helped shape the question better. In an interactive site it is natural posts are refined as points come up. As long as reasonable time is given, one cannot be 24/7 online.
    – Nikos M.
    Commented Jul 23, 2022 at 13:11

It depends on how far away from “fair coins” these coins are, if all we care about is whether “heads” or “tails” is thrown.

Assume the k-th coin comes up heads with propability p_k, and the average value of p_k for all coins is p.

Throwing n coins, picking a random coin for each throw, will give np heads, plus/minus some statistical variation. Picking one coin and throwing it n times will give np_k heads, plus some statistical variation.

Depending on which coin you picked (whether p_k is close to p or not), how many coins you throw, and how far that statistical variation took you, your outcome will be different depending on how you play.

How far the p_k are from each other is important. If you have various p_k between 0.4 and 0.6 you will find out quickly, if they are from 0.499 to 0.501 or even from 0.426 to 0.428 it will be harder.

If you have mathematically perfect coins then the results are indistinguishable.

  • Please see updated question, you have all the time to update your answer too.
    – Nikos M.
    Commented Jul 20, 2022 at 10:20

In the first experiment, throwing the same coin N times and getting an outcome of all heads is increasingly problematic for the same fair coin.

You've condensed the list of possible issues (i.e. cases where you're majorly off the statistically expected outcome) into "the coin is not fair", which inherently skews your perspective towards "we can only meaningfully detect issues when using the same coin all the time"

Whereas for the second experiment, getting an outcome of all heads, is not problematic for any coin, simply thrown once, thus not problematic for the whole outcome.

I very much disagree that there is no indication of a problem here.

I'm going to use 50 here as an arbitrary value.

If you consider the same coin landing heads 50 times in a row indicative of a problem, then you should consider 50 coins all landing heads at the same time indicative of a problem as well.

Sure, the explanation of the issue can't be "this specific coin is unfair" since there is no "this specific coin" in this case, but there's a whole range of possible issues that can still be identified:

  • Coins are systematically unfair, e.g. because the head shape takes more material, therefore impacting the odds of landing heads vs tails. Since this has to do with the design of a coin, not a specific coin, it can be identified by flipping many coins (and cannot be identified by flipping a single coin)
  • The flipper is eerily consistent about their method of flipping, therefore being able to consistently achieve the same outcome. (Whether true or not, I've been told in the past that whatever side is facing up when you start flipping is slightly more likely to be the outcome, at a margin of about 51-49)
  • The environment in which the coins are being flipped somehow impacts its trajectory and ends up favoring it landing on one spot (e.g. a coin with one magnetic and one non-magnetic side would receive interference from a magnetic landing zone).

This is by no means an exhaustive list.

The issue isn't that flipping individual coins cannot highlight a problem. The issue is that you're only considering a problem that inherently entails one specific coin being the cause of the problem.

  1. In the first experiment, throwing the same coin N times and getting an outcome of all heads is increasingly problematic for the same fair coin.

This is not necessarily problematic, as their is nothing particularly special about the all heads outcome, which has the same low probability as any other particular sequence of heads and tails.

  1. Whereas for the second experiment, getting an outcome of all heads, is not problematic for any coin, simply thrown once, thus not problematic for the whole outcome.

This is a fallacy of composition. The fact that no individual coin might be problematic does not necessarily mean that the ensemble of coins is not problematic.

  1. If we say that the first experiment is the same in this sense, that N throws is simply a series of one throw, then either throwing N times looses meaning, or by same token as point 2) we have no reason to infer the coin is not fair by observing any long sequence of heads, which seems paradoxical.

This relies on point 2 being valid, which it isn't. Thus, there is nothing paradoxical.

  1. If the sequence is all that matters and not the coins themselves, then we can infer in second experiment that all coins are not fair, simply by one throw of each, which seems paradoxical.

This is a fallacy of division. Just because the ensemble of coins might not be fair does not make any individual coin unfair.

  1. if the sequence is all that matters and different coins can really be anywhere at anytime simply thrown once, this raises questions how the different coins are synchronized so that the sequence satisfies certain statistical properties which are similar to those of one and same coin which seems paradoxical, whereas the same questions are not necessarily raised for one and the same coin.

I'm not entirely sure what you're trying to say here or what "how the different coins are synchronized" means, but I think what you've said here is addressed by what I've said above.

I think the main thing to take away is that, while one can obviously tell the difference between physically flipping one coin many times and flipping many coins one time, both scenarios can be mathematically modelled by the same random variable, so perhaps it might be better to use a word like "isomorphic," rather than "identical."

  • The fallacies you mention are not always fallacies, so one has to reason based on something else that they are indeed fallacies.
    – Nikos M.
    Commented Jul 21, 2022 at 7:51

A "fair" coin is an idealization as is perfect order and perfect disorder (randomness). A truly fair coin would be impossible since the coin and its surfaces would have to be identical thus making it impossible to distinguish between the two sides. Simply making the two surfaces identifiable guarantees that a real coin is unfair and inconsistent from coin to coin.

Your random experiments are different and have an extremely high probabilty of producing different outcomes. No two real coins will ever have exactly the same probability.

So how do I determine "fairness" under non-ideal conditions? Using the ideal as the standard, attempt to make coins as close to 50/50 as possible and perform experiments to determine the actual probabilities. For example:

A prototype coin has pH of 50.0023 after 10,000 tosses. Is this fair enough? It depends on how sensitive your system is to non randomness. If it's acceptable, then how repeatable the manufacturing process is will determine the uncertainty in this probability.

So if your coins have a given probability and uncertainties for N tosses, and these are acceptable as "fair" for your system, then your experiments can be treated as if they are identical even though they are not.


Short Answer

The long answer is that most mathematicians would consider them identical in the sense they endorse the notion of independent events. A sequence of coin tosses of the same coin n times is in terms of probability is the same as n coins tossed if one takes the a sequence of physical locations to build a sequence out of the second case. What you are digging at is why.

Long Answer

So, a philosophical analysis would revolve around your vocabulary use. To you, it's "problematic" and "unproblematic" when one is dealing with a sequence of heads versus when one is dealing with a collection of coins tossed simultaneously arranged in a sequence. To wit:

In the first experiment, throwing the same coin N times and getting an outcome of all heads is increasingly problematic for the same fair coin... Whereas for the second experiment, getting an outcome of all heads, is not problematic for any coin, simply thrown once, thus not problematic for the whole outcome.

Your first statement is the gambler's fallacy and the second statement is the fallacy of composition. So, let me say a few words first, to reorient the conversation.

First, "problematic" has no definition. As a debater, I would mock for you using this term in mathematical discourse to get laughs. But here, my goal is not to win, but to persuade you that "problematic" does not exist in a math dictionary in any sense connected to frequentist notions of independence. (Of course, feel free to produce a counterexample.) See, were this a debate, the argument is already 80% over, because your first two statements are well-known fallacies. But, since this isn't a debate, let's explicate on what is happening with your thinking.

Idiosyncratic Terminology

Your term "problematic" is a red flag. Problematic is a very ambiguous, lay term that expresses normativity without a framework. If mathematicians used the term "problematic", there would be a rigorous definition argued over the community and available. Do a search for the term "problematic" in the WP article on independent events. Know how many hits? Zero, zilch, zip, none, bupkis. That's a clue that you're dealing with idiosyncratic idiolect or vagueness or novelty leading up to a neologism. If problematic has no generally accepted mathematical usage as jargon, what should be made of it?

I would simply argue you are using your intuition, and doing so reasonably. Some philosophers hate intuition. That want deductive certainty. But deductive certainty or worse, certainty by faith applied to real-life problems is a poor choice. That's not to say intuitions are always right. Certainly not. I once went to a riverboat with a pet "theory" based on an intuition, that doubling a bet would be a good strategy at roulette. My reason was the same as yours. It would be increasingly unlikely for the next flip to come out red after a long sequences of black. Mein was a Gedankspiel, so I simply wagered in my head for an hour. I was overwhelmingly in the negative based on my intuitions of probabilities of sequences. And that's not because I'm stupid, but because my brain, like most brains faces cognitive biases. Simply put, there is a psychology to misrepresentations like fallacy. The WP article on the gambler's fallacy has an entire section devoted to it.

Cognitive Biases, Fallacies, and Statistics

Yes, our brains are hardwired to think just as you claim, that increasingly long sequences of the same or similar values influence continued results. But it's an illusion, and here's why. on the one hand, you can view a sequence of n tosses as 1/2^n which are very special. They're all heads! Whereas the remainder of non-all heads sequences is 2^n-1 and are not special patterns. So it seems odd to have 1/2^n then 1/2^(n+1) and then 1/2^(n+2) all happen in a row. Doesn't feel right to have three rare events happen in a row, does it. But there's another interpretation.

The laws of physics simply don't care (on a fair coin) what is a head or tail. There are only two sides, and flip and come what may. You see, physics is blind to whether it is heads or tails. A fair coin flipped in sequence is just randomly determined by stochastic processes, and by definition they cannot be affected by the orientation of the coin upon landing. The final orientation is incidental the degrees of freedom which govern the motion. There's the hand flip, the wind, the Coriolis effect, resistance to the wind, variations of the distance from the center of Earth's gravity, etc. But BY DEFINITION, the orientation of the coin has no role in shaping the motion because the orientation of a coin isn't a physical property, it is a mental one of the observer.

Let's try another example. Fill a plastic bottle with 50 red marbles and then with 50 green ones on top. Two layers. Shake. And repeat. And repeat. And as long as you are alive, the odds of you shaking them back into two layers are so theoretically minuscule, that with confidence one can say it won't happen. This is the nature of entropy. Why? Because the color of the marble, isn't a physical property, strictly speaking. Colors are constructed by our bodies and brains, at least according to theories of embodied cognition. To a red-green blind person, there isn't even an experiment here to be had. And the universe is no person at all. In the same way, when you flip a fair coin, the randomness that inheres in the universe conducts that coin into a set of frequencies that can be demonstrated empirically over and over.

When I attended a university once long ago, our teacher had us in pairs create a a really long sequence of random numbers with our intuitions, and then with a coin. The cognitive bias towards our intuitions is so strong, he plugged the sequences into software and could tell for almost every group in the class, which of ours was done with a coin, and which we made up. It seems a fantstic claim, but it's actually one of the central theses of Thinking, Fast and Slow. Our brains are just really bad at statistics, and as Kahneman points out, his research demonstrates that extends to professional statisticians! (The man won a fancy prize, and the book is worth a read.)

A Closer Look at the Second Fallacy

getting an outcome of all heads, is not problematic for any coin, simply thrown once, thus not problematic for the whole...

From WP:

The fallacy of composition is an informal fallacy that arises when one infers that something is true of the whole from the fact that it is true of some part of the whole.

You almost nailed it word for word. ;) What's the argument here? In fact, there's an easy counterargument from math. Measures and measures of central tendency are not the same thing. But yet, we again are capable of biases that mislead us. If I tell you that a set of integers has a mean, median, and mode of 50, and that the values go from 1 to 100, what can we infer about the underlying set? Not much. But our minds will try to fill that set. We can have ten values of 50. We can have a set with an outlier, and a skew on the other side to offset it. We can a sequence leading to 49, 2 50's and a sequence starting at 51. We simply don't know, and our brains our evolved to know. So the brain takes a guess, or fills in the blanks, or sees a picture of your favorite deity in your breakfast toast.


The gambler's fallacy is a well established empirical fact. And event independence is also a well established mathematical and empirical fact. And our logic, when we become specious, is often lead astray by our intuitions. Casinos make billions, and the lotteries are called taxes on the mathematically illiterate because the findings of statistics are pretty dang reliable. What you need to ask yourself is why do you think you can out argue hundreds of years of peer-reviewed, empirically validated statistical research? Why do you believe that intuitions that arise from your cognitive dissonance somehow invalidate the categories of fallacy created and accepted by logicians all over the world? When you are confronted with the fact that your words contradict (a sequence of fair coin toss by definition is random, and your knowledge of it doesn't impact that sequence), why do you scamper to find better words to affirm your feeling of distrust of the claim? That's easy! It's human nature, and we all do. Some of us just learn to spot the misrepresentations like the gambler's fallacy, the fallacy of composition, hasty generalization, and so on. Trust me, if longer and longer sequences were "problematic", I would have cleaned out the casino at the roulette table long ago. ;)

  • I nailed the compositional fallacy word for word, that is for sure. But the fallacy is only informal, it is not always fallacy. Pointing at other (mathematical) examples where it is a fallacy, does not necessarily make it a fallacy in this case. This is a major point of the question.
    – Nikos M.
    Commented Jul 22, 2022 at 8:42
  • For example saying that each part of my phone is made of electrons,protons,neutrons and energy, thus my whole phone is made of electrons,protons,neutrons and energy
    – Nikos M.
    Commented Jul 22, 2022 at 8:59
  • Exactly. So the burden of proof now falls on you to show why either claim is still valid. You can attempt to defend both claims. If you refine either points about sequences or single throws as a group, then the argument doesn't die. Perhaps by problematic means "empirically implausible" in that case, you'd provide an argument as to why. If there is a reason why v what c is true of the parts is true of the whole, you'd have to go into detail as to b why. In essence, you have two new areas of inquiry. Is the each point a fallacy or not? It could be worth additional clarification ..
    – J D
    Commented Jul 22, 2022 at 12:02
  • Perhaps when you say in the "literature", you can provide a reference where you can show a weak argument about sequences. Your question is about mathematical conclusions, but there is no mathematical counterarguments...
    – J D
    Commented Jul 22, 2022 at 12:05
  • Your best hope is to defend the notion of problematic. It's a weasel word. How exactly is it "problematic to get a long sequence of heads"? If you mean every throw decreases the odds of the next flip being a head as the sequence grows, then you have to explain exactly why the coin flip isn't 50-50 anymore which is the standard view. And you have to reconcile that with your acceptance that is a fair coin, bc those claims would be contradictory.
    – J D
    Commented Jul 22, 2022 at 12:13

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