Are these random experiments the same?

Question

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

Answer 1

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.

Answer 2

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?

Answer 3

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.

Answer 4

"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.

Answer 5

Caveat

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.

Conclusion

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.

Answer 6

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.

Answer 7

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.

Answer 8

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.

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.

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.

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.

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

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

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.

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."

Answer 9

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.

Answer 10

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.

Conclusion

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. ;)