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Does chance actually exist for a single case? Even for a coin, what does it mean to say that there is a 50% chance that the next coin toss will land on heads?

Someone might say that this means that if one were to throw this coin an infinite number of times, it’ll land on heads half the time. But this is a property of a long running series of tosses. And this is atleast somewhat testable with respect to a large series of tosses. One can throw it many times and simply calculate the frequency.

How does one test that the next coin toss has a % chance of landing on a side? Is there a property or concept in reality where the chance of the next single coin toss or your next dart throw hitting the bulls eye actually means something?

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    case chance is not about existence, it is about probabilities. Existence it is that a coin have 2 sides, it is not probability, you cant make coin without 1 side or with 3 sides. Existence is predicate to probability logic. Coin is exist that mean that it has 2 sides, and 2 sides sign that chance is 50/50 on next toss. If you expand existence(add pseudo existing) you ll get new probability function, if you have second toss and you know that first was a-side, then you ll have 25/75 to a-side getting next toss. Existence of the condition determines the probability function you should to use. Mar 5, 2023 at 8:26
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    It does on propensity interpretations of probability. This is often applied in quantum mechanics: an electron has equal dispositions to go through either slit in the double slit experiment. This manifests as going through each slit with 50% frequency in a long series of runs, but the disposition is there on each run, or even if no runs are performed at all. We may not be able to detect such disposition without a series of runs, but our inability to detect something in a particular way does not mean much. We are only human.
    – Conifold
    Mar 5, 2023 at 9:56
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    @thinkingman Science can get by and has with arguments from best explanation. Unfasifiability is not the only metric for theories. Just yesterday I heard Huw Price say this about backwards causation in QM as a possible future theory choice. Many theories of QM might be unfalsiafiable, yet he and physicists aren't stopping work in finding the best one with rigorous work.
    – J Kusin
    Mar 5, 2023 at 17:24
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    The interpretation of probability you give in your question is the frequency interpretation. There are many more interpretations.
    – Stef
    Mar 5, 2023 at 20:11
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    Consider the following experiment: I flip a coin, I look at the result, but I hide it from you. Then I ask you, what is the probability that the coin has landed on heads? You have to say 50%, even though the coin has already landed.
    – Stef
    Mar 5, 2023 at 20:19

8 Answers 8

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You've hit upon a frequently debated topic in statistics, that is, what does "probability" actually mean?

At the moment, the philosophical arguments tend to boil down to two main camps: Frequentist and Bayesian.

The Bayesian perspective is that probability reflects a substantiated degree of belief about the outcome(s). Note that this isn't just a "gut instinct" about things, but instead comes from a systematic combination of all the relevant factors. That is, if you're calculating the probability of heads, you start with your knowledge of the physics of coin flipping, add on the observed results of all the prior coin flips you're aware of, combine in your understanding of how many double-headed coins exist, if the person flipping supplied the coin, if he has money riding on the outcome, etc. You combine all these factors together in a defined way to get the probability for that particular coin flip.

That probability number represents your belief as to the outcome of the flip. Note, though, that someone else with other information (e.g. who has done a more in-depth study of double-headed coins, or who is aware that the person who is flipping is a slight-of-hand artist) might come up with a different probability. For a Bayesian perspective, this isn't an issue. Probability quantifies beliefs, so if someone has different beliefs, they'll have different probabilities. The Bayesian system is set up, though, such that as you input more and more information, you should converge more and more accurately to the observed frequencies of events.

The Frequentist approach, on the other hand, rejects the notion that individual events have probability. A probability is an average over a (large) number of related events. So sensu stricto, one cannot talk about the probability of an individual event. However, it's a common shorthand to take the "probability" of a single event of unknown outcome to be the probability of the set of events of which that event is a class. (As the probability of a single event is otherwise nonsensical.) So "the probability this coin lands heads" is less "what is the probability that this particular 1998 US penny from the Denver mint flipped by thinkingman at exactly 15:43:23 UTC 2023-03-05" but rather "for the set of government-issued coins flipped by arbitrary humans at arbitrary times, what is the probability of a head?"

This implicit sliding between "probability of a specific event" and "probability of a set of events, of which this event is a member" can get confusing. It's particularly fraught when discussing events for which no replicates are possible. For example, the probability for [insert politician] to win the 2024 US presidential election. The 2024 presidential election is a single event which is impossible to replicate. What does it mean to have a frequentist probability of something which will only ever have one replicate? In these cases, you're talking about a set of hypothetical events, each one which has all the same important characteristics of the event in question. It's not really a probability for a single event, it's the probability we would get for the hypothetical set, if we hypothetically were somehow able to repeat the event multiple times.

This "set of replicates" viewpoint depends critically on which parameters you keep constant. We can restrict the set of relevant events to those involving US pennies. We can restrict thing to only known fair coins. We can restrict it cases where the flipper is a known slight-of-hand artist. A frequentist can come up with different probabilities for the same event, not because the beliefs are different, but because the relevant set of replicates is potentially different. That doesn't really affect the frequentist perspective, as again, specific events don't really have a probability, it's a set of events (of a particular type) which do.


As a final note, while often phrased akin to religious affiliations, in practice "Frequentist" and "Bayesian" should be more viewed as perspectives. It's not a "one is right"/"one is wrong" situation, and most statisticians will switch back and forth between the perspectives depending on what they want to accomplish.

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    I disagree that "frequentist" and "Bayesian" are "the two main camps" in the debate about interpretation of probabilities. Those two are approaches, not interpretations. They are the two main paradigms that guide the reasoning, the proofs, the calculations, when solving a problem with probabilities. But whether a mathematical proof follows a Bayesian or frequentist approach doesn't really say anything about how to interpret those probabilities once they have been calculated.
    – Stef
    Mar 5, 2023 at 20:15
  • I agree with your last paragraph, neither of the two approaches is wrong - this is math we are talking about, after all, not politics or religion. However, I would strongly advise against switching back and forth between the two approaches when in the middle of a proof or mathematical argument, as it's a guaranteed way of making mistakes in the reasoning, and inevitably getting a wrong result.
    – Stef
    Mar 5, 2023 at 20:17
  • I am not so sure about your example of the 2024 US presidential election, as the outcome will be determined by millions of individual events, and each potential voter's disposition to vote for a particular candidate (or whether to vote at all) is not a once-only event. If I were to pick an example of something where assigning any probability seems problematical, I would go with the question posed by the title of Bostrom's (in)famous paper, "Are You Living in a Computer Simulation?"
    – sdenham
    Mar 9, 2023 at 17:49
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You are missing the point that probability and statistics, as physics, provide models for reality, they do not say anything about the actual nature of reality, but just function as different formalizations of intuitions that are behind how the world operates. That's precisely why models are discarded when new evidence or reasoning provide basis for their obsolescence, and they're replace by new or tweaked models.

Therefore it's incorrect to ask if there is anything in reality that corresponds to the 50% chance defined by probability. The first phrase that out Probability professor said when using the coin example was always "Imagine we are performing a perfect, isolated experiment with a perfectly crafted coin, then the random variable...". As you can see, this setup does not correspond with anything in reality, but it's an useful model nevertheless.

Unless you follow some kind of platonism of course, in that case you could say that every mathematical object is some kind of Form. But strictly speaking chance is a mathematical construction to help modelling our empirical experience of randomness, should it exist.

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  • The problem is that physical equations can be measured. There is no debate between what the value of gravity is. There is a “real” value of gravity in actuality even if we use a human form of language known as math to describe it. There is no “real” value for the probability Joe will get cancer tomorrow, since the very notion of that is subjective. Mar 5, 2023 at 20:38
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    @thinkingman The value for gravity also comes from a mathematical model applied in physics to model interaction of forces between two bodies. It has no real value whatsoever, because it corresponds to a simplified abstraction of reality, as happens with chance. There is indeed debate about the model to use for modelling such interactions, see Gravity, alternative modern theories.
    – user64708
    Mar 5, 2023 at 21:38
  • We have physical laws because experience has told us that apples falling from a tree have similar behaviour. With chance it's the same, because experience also shows us what happened before, then we are able to model single cases. In physics you are modelling the fall of a new apple without taking into account that it's an unique apple, different from the others that fell before. But maybe this particular apple is empty because a worm ate it after you took your measurements, and thus your model will fail, as also can happen with chance.
    – user64708
    Mar 5, 2023 at 21:45
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    Neither concepts, gravity or chance, are real, unless you follow some kind of platonism. They're mathematical constructions that help model our empiric experience.
    – user64708
    Mar 5, 2023 at 21:46
  • Gravity is real in that the concept maps to an actual force that exists in reality. Probability does not Mar 5, 2023 at 23:49
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You asked two linked questions:-

  1. "Even for a coin, what does it mean to say that there is a 50% chance that the next coin toss will land on heads?"

What it means to say that there is a 50% chance that the next coin will land on heads is that there are by definition just two outcomes and one of them must be true.

  1. "How does one test that the next coin toss has a % chance of landing on a side?"

The only empirical test is to actually toss the coin a large number of times. You will find out that the empirical probability is unlikely to be exactly 50%. You may also find out about all, or at least some, of the ways that the rule can be violated as well.

The problem is that one doesn't, or rather can't, empirically test a probability like this one, because it isn't an empirical prediction. It follows from the rules of the game and is in that sense analytic.

If you were to choose a different kind of probability, the answer would very likely be different.

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  • "What it means to say that there is a 50% chance that the next coin will land on heads is that there are by definition just two outcomes and one of them must be true." If the question was "Will I get hit by lightning tomorrow?" there are also by definition two outcomes, and one of them must be true, but I certainly hope the probability is not 50%.
    – Stef
    Mar 6, 2023 at 12:45
  • Good point. Needs thinking about. The difference is that the chance is not part of a game with rigidly defined outcomes. You would have to appeal to empirical statistics. That's a different kind of probability and, I think, "the chance of you being struck by lightning tomorrow is ****"
    – Ludwig V
    Mar 6, 2023 at 13:08
  • A game with rigidly defined outcomes: I draw a card out of a deck of 52 cards. The question is "Do I draw the ace of spades?". There are by definition two outcomes, "yes" and "no", yet one of these two outcomes has probability 1/52 and the other outcome has probability 51/52.
    – Stef
    Mar 6, 2023 at 13:10
  • That's valid. But from another point of view, there are 52 outcomes. Your binary choice misrepresents the game.
    – Ludwig V
    Mar 6, 2023 at 13:13
  • Okay. New experiment. I have an unspecified number of decks of 52 cards, which I all mix together. I draw a card from this mixed deck. The question is still "Do I draw the ace of spades?". How many outcomes are there? I would say there are still only two possible outcomes, either I draw the ace of spades or I don't, and the probability that I draw the ace of spades is still 1/52, regardless of the total number of cards. And I don't think I'm misrepresenting the game.
    – Stef
    Mar 6, 2023 at 13:17
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When you are about to toss a coin, you cannot tell how it will land. You know it must land in one of two ways. You have no reason to suppose it will land in one particular way. Given that, you conclude that it could land either way, with the circumstances not favouring one or the other. That is all you mean when you say there is a 50% chance the toss will land on heads. Now, given that, why are you perplexed about it?

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I do not like the example of a coin.

How about an explosion or a big bang or some kind of creation of the universe ?

If it can not be repeated then does probability have a meaning ?

If we have full information about what happened and when then I say probability is meaningless if the event can not happen again.

However we might be able to compute the probability of the event happening at that point in time or space based upon a model.

That does ofcourse assume the model is correct, what is difficult if the event can not be repeated.

In short :

No in my opinion.

Notice non-repeatable is pretty close to unfalsifiable.

But there is another thing.

What is the probability that you have the name that you do ?

That you are alive in this century and not another ?

That the universe or life exists ?

You see we can pick proven or tautological facts and argue its probability is not 1 but when something is true it happened with probability 1 in a way.

Because if we talk all facts as low probability then nothing exists statistically ! That would be absurd.

A better question would be, what is the probability that someone asked the same question as you in chinese at the same day ?

But then you got a repetition.

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In the single case context chance does not refer to the probability of the result.

If the coin lands heads up by chance, that means that the result is random, no-one has intentionally selected it.

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First, I would like to move from coins to dice. I just want to move away from 50:50 because that can cause confusion.

Now, what does it mean to say that a standard die is unfair? It means that the probability of it landing on each face is not equal, but that doesn't much help determine what probability is!

Let's go with a propensity model for illustration, which is to say "there is something causal about this die which makes it more likely to come up with a 4." That could be that some of the corners have been clipped, there's a weight on one side, there's something magnetic, whatever. It's a quantifiable property of the die. You could, even without throwing the die, probe it with calipers, scales, and other tools to predict the long run probability distribution. Now since it is a property of the die, and presuming that whatever that property is it remains stable, it seems natural to say that there is a bias to the probability distribution on a single throw, just as much as there is one in repeat.

Still, working out what that property is remains delicate. Throwing a die is actually a deterministic process, which raises questions about what we mean. Essentially the property of unfairness is that, over all the possible initial positions and momentums with which the die may be thrown, more than one sixth of them would result in a 4.

Here you may argue that for a specific throw, we don't get to integrate over all initial orientations but get one. It is no good stepping back to the fairness or otherwise of the shaker; if the universe is relevantly deterministic it is also just a device mapping a fixed input orientation and some arm jiggling to position and spin in the dice, and the human is a device for mapping a complex array of psychological and biological factors to arm jiggling. So sure enough, for that particular throw single case chance is 1 for one outcome and 0 for everything else.

There is a catch though. Assuming a deterministic universe, the exact same thing holds for long run chance too. The exact sequence 2, 4, 5, 4, 4, 2, 4, 1, 4 was exactly the sequence which had to happen, and all others didn't get the right initial conditions. Still, we find that we can meaningfully talk about long run probability, whether because we aren't determinists or because we are talking about something different like epistemic probability.

So, what is single case chance? It's whatever long run chance is, just shorter. If chance is shorthand for integrating over undetermined aspects of reality, that could apply to the single case. If it is shorthand for integrating over unknowns, there are plenty of unknowns in the single case. If it just a useful mental tool for deciding on behaviour, empirically people bet on one offs all the time. The only difference is that, barring a way to explore some hypothetical multiverse, we can't check the probability distribution of one offs was what we expected.

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How does one test that the next coin toss has a % chance of landing on a side?

Toss the coin ten times and record the results. You can use this to estimate the probability that the next toss will be heads or tails. It will just be an estimate, but it will be a meaningful estimate. Just try it—you'll find your estimate works much better than if you just picked a random percent chance for heads or tails.

Every single thing ever measured in physics is like this. Some things we have incredibly, miraculously precise and accurate measurements of, but they're still estimates. You can never measure anything with true, perfect accuracy in the physical world—you can only pretend to (that's the mathematics).

Is there a property or concept in reality where the chance of the next single coin toss or your next dart throw hitting the bulls eye actually means something?

You bet. Try it—toss the coin ten times or throw the dart ten times. If you're not satisfied at that point, do another ten, and you'll get a better measurement. You'll see how real it is. It may not be perfect (it can never be perfect) but it will be much better than arbitrary, and that does mean something.

EDIT: In a comment, Ludwig V pointed out that this answer is based on a discussion of multiple events, whereas the original question is based on the chance of a single case, implying that I take the supposition of a single case as somehow flawed. That's true. They also asked me to explain my reasoning, so I will.

With a question like this, I think it really helps to think in pragmatic terms—as in, what is the utility of assigning a probability to something like the outcome of a coin toss? I'd say the most obvious, fundamental motiviation for doing that is to try to predict it. Saying that a coin toss has a 50% chance of heads or tails, if true, is appealing because it tells you that you have no reason to prefer heads or tails, like if you're betting on the outcome. People make use of this idea frequently in daily life.

In line with this, when someone says that a "single coin toss" has a 50% chance to be heads or tails, they are implicitly making an educated guess about the result of the coin toss. Hopefully you agree this far. For them to truly make an educated guess, it implies that the "single coin toss" is not actually single at all—rather, implicit in their guess is that the coin toss in question has many things in common with all the other coin tosses the person has observed. People are comfortable making guesses like this because the outcome of most coin tosses in daily life does at least appear to be reliably split between heads and tails, at least for practical purposes.

All that happens when you perform the next coin toss, or throw the next dart, or what have you, is that you add another sample to the pool of all the coin tosses or dart throws you've observed in your life, which may slightly refine your overall sense of the probable outcomes of those events. To calculate the chance of the coin landing on its side in the next toss, all you need is a record of coin tosses; you can see how many times the coin landed on its side and extrapolate from there. Your guess may not be perfect—again, it can't be perfect, as we can't measure anything in the physical world with perfect accuracy—but given that coin tosses tend to resemble each other, it will probably be a better guess than if you guessed randomly. Just because the new event is unique in some sense doesn't mean it has no resemblance to other past events of its kind.

Overall, I think it's important to note than neither a coin toss nor a dart throw can ever truly be a "single event" with no relevance to other coin tosses or dart throws. Obviously each one is unique in some sense, but they must have some resemblance to other similar events or there would be no way to make reliable predictions about their outcomes. If we knew that a truly unique event was going to occur, with no resemblance to anything that has ever happened that we know of, I think it's fair to say that we could not reasonably make any predictions about it, but such an event, by definition, can't even be imagined any further than that.

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  • Welcome to SE. This is an answer to the question, but a rather strange one. You ignore the presupposition of the question - something that is assumed in asking the question. (A good example is the lawyer who asks the accused "Have you stopped removing goods from shops?") The presupposition of this question is that the test you describe (and which is described in the question) does not give an answer, because it involves multiple tosses of the coin. You ignore that, so effectively you are saying that presupposition is wrong. You need to say so and explain why you believe that.
    – Ludwig V
    Mar 7, 2023 at 18:59
  • @LudwigV All right, will do. Mar 10, 2023 at 21:48

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