B How might determinism affect probability?

[cliffhanley203]

The probability, I was taught, for red to appear at roulette (on a European table, with a single green zero) is 18/37; and the probability for non-red to appear (black or green zero) is 19/37. If we live in a deterministic universe (I appreciate that that’s a big if for some, not so much for others) is the actual probabilites for any given spin one and zero (1 and zero) e.g, if it’s been determined that red 36 will appear then it’s 100% guaranteed that red 36 will appear (and 100% guaranteed that non-red will appear. So, in that situation, is the probability for red 1, and the probability of non-red 0?

P.s. I appreciate that, unless you were Laplace’s demon, you couldn’t know this in advance of the spin but I’m still wonderin if, even though we would describe the probabilites as 18/37 and 19/37, they would actually be 1 and 0.

 

[member 587159]

This is rather a philosophical question, and it doesn't make sense to me to talk about probability in a deterministic context.

 

[Dale]

There are at least two different schools of thought regarding probabilities.

One is the “frequentists” which view probability similarly to how you are suggesting. Probabilities only exist as the fraction of results with a given outcome after an infinite number of trials. So probability is not defined for a single spin. You would have to spin an infinite number of times and get red each time for the probability to be 1.

The other is the “Bayesians” which take probability as a measure of our belief. For them probability of a single event is well defined, and can even be measured in terms of hypothetical bets. Even if the universe is truly deterministic, we may not know enough to determine the outcome, so the probability would account for our ignorance.

 

[cliffhanley203]

This is rather a philosophical question, and it doesn't make sense to me to talk about probability in a deterministic context.

I'm not sure if the determinism part is only philosophical. I've spoken to scientists who see determinism as the 'only game in town' due to their belief in causation (although some scientists I've spoken to believe quantum indeterminacy means we can't be certain of determinism). So for some it seems to be a physics question as well as a philosophical one. It seems to me to be a maths question also. If I asked what is the probability of non-red appearing on a wheel with no non-red pockets my guess is that most mathematicians would say it's zero. The above roulette wheel (in the universe where it's determined that red will appear) is analogous to the wheel with no non-red pockets, which is why I'm wondering if the probability is also zero.

 

[cliffhanley203]

There are at least two different schools of thought regarding probabilities.

One is the “frequentists” which view probability similarly to how you are suggesting. Probabilities only exist as the fraction of results with a given outcome after an infinite number of trials. So probability is not defined for a single spin. You would have to spin an infinite number of times and get red each time for the probability to be 1.

The other is the “Bayesians” which take probability as a measure of our belief. For them probability of a single event is well defined, and can even be measured in terms of hypothetical bets. Even if the universe is truly deterministic, we may not know enough to determine the outcome, so the probability would account for our ignorance.

“One is the “frequentists”...”

Which of these two schools of thought dominate maths (I’m thinking of your average textbook which tells us that the P of rolling a 6 on a die is 1/6 etc)?

“...after an infinite number of trials.”

Is this the idea that we will see exactly 18/37 reds and 19/37 non-reds over an infinite number of trials? Or exactly 50% heads, 50% tails etc?

“So probability is not defined for a single spin.”

So it’s not the frequentists who are writing your average textbook?

“You would have to spin an infinite number of times and get red each time for the probability to be 1.”

Which can’t happen, even in theory, given that there’s 18 red pockets, 19 non-red pockets, yeah?

“For them (the Bayesians) probability of a single event is well defined, and can even be measured in terms of hypothetical bets.”

What would a Bayesian say about a single spin of the wheel regards red v non-red?

 

[BWV]

There is no question that the physics of a roulette ball are completely deterministic - if you know precisely all the relevant variables you can predict the outcome with 100% accuracy. But, because the outcome is extremely sensitive to initial conditions - the force the ball is dropped, the spin rate of the wheel, where the ball lands on the wheel etc, it is effectively random and the use of probability reflects our ignorance of the outcome - i.e. in the absence of any relevant information, any particular spot on the wheel is as likely as any other.

 

[Stephen Tashi]

Which of these two schools of thought dominate maths (I’m thinking of your average textbook which tells us that the P of rolling a 6 on a die is 1/6 etc)?

We must distinguish between mathematics and the application of mathematics. The mathematics of probability theory makes no comment of how physical situations produce probabilities. It doesn't specify any relation between probabilities and the actual frequencies with which events occur. If a math textbook states something about the probability of rolling a 6 with a "fair" die, it is merely a hypothesis in a problem - in the same sense that a statement like "Ed has 6 apples" can be a hypothesis (a "given") in algebra problem. As far as the pure mathematics of probability theory goes, it is a special case of "measure theory". There is no distinct "frequentist measure theory" versus "Bayesian measure theory".

It is in applications of probability theory that there is a distinction between the Bayesian and frequentist approaches. To get mathematical solutions to real life problems, assumptions must be made. The frequentist and Bayesian approaches differ in how they choose to make assumptions. When it comes to applying probability theory to physics, there are many different interpretations of probability. In fact, there are many different interpretations of probability among people who call themselves Bayesians.

Textbooks on probability theory often try to be helpful by mixing the mathematics of probability with examples of its application. For example, the typical text on introductory statistics teaches the "frequentist" approach to applying probability theory. Modern textbooks about probability theory (as opposed being only about the topic of statistics) usually show both the frequentist and Bayesian approaches.

 

[Stephen Tashi]

There is no question that the physics of a roulette ball are completely deterministic

Of course, you mean the Newtonian physics.

 

[Mark44]

P.s. I appreciate that, unless you were Laplace’s demon, you couldn’t know this in advance of the spin but I’m still wonderin if, even though we would describe the probabilites as 18/37 and 19/37, they would actually be 1 and 0.

I'm thinking that maybe you don't understand probability.

“One is the “frequentists”...”

This isn't a term that mathematicians use to describe themselves, to the best of my knowledge.

Which of these two schools of thought dominate maths (I’m thinking of your average textbook which tells us that the P of rolling a 6 on a die is 1/6 etc)?

Are you questioning whether this probability is correct?

 

[Dale]

Which of these two schools of thought dominate maths (I’m thinking of your average textbook which tells us that the P of rolling a 6 on a die is 1/6 etc)?

The frequentist school of thought is by far more common. Although both will give you the same p for rolling a fair die.

Is this the idea that we will see exactly 18/37 reds and 19/37 non-reds over an infinite number of trials? Or exactly 50% heads, 50% tails etc?

Yes, assuming tautologically that it is a fair wheel or a fair coin.

So it’s not the frequentists who are writing your average textbook?

It is the frequentists, but the question you asked is not something that is addressed in standard courses.

What would a Bayesian say about a single spin of the wheel regards red v non-red?

That depends on what prior information they have. If they know with complete certainty the determined outcome then they would say 1 or 0 accordingly. Otherwise they would take any limited information they have into account.

 

[cliffhanley203]

There is no question that the physics of a roulette ball are completely deterministic - if you know precisely all the relevant variables you can predict the outcome with 100% accuracy. But, because the outcome is extremely sensitive to initial conditions - the force the ball is dropped, the spin rate of the wheel, where the ball lands on the wheel etc, it is effectively random and the use of probability reflects our ignorance of the outcome - i.e. in the absence of any relevant information, any particular spot on the wheel is as likely as any other.

“...the use of probability reflects our ignorance of the outcome - i.e. in the absence of any relevant information, any particular spot on the wheel is as likely as any other.”

I appreciate that assuming it is deterministic (as you say it is) we are still ignorant of what the outcome will be; but I don’t see how “any particular spot on the wheel is as likely as any other” if it is deterministic. I can see why from our subjective point of view it would seem like that, given our ignorance of the outcome, but the objective reality would be that THERE IS ONLY ONE POSSIBLE OUTCOME, i.e, the ball will land in a red pocket because that is what has been determined by antecedent events. Which is why I’m wondering if the actual probability for red is 1, and the actual probability for non-red is 0.

 

[cliffhanley203]

We must distinguish between mathematics and the application of mathematics. The mathematics of probability theory makes no comment of how physical situations produce probabilities. It doesn't specify any relation between probabilities and the actual frequencies with which events occur. If a math textbook states something about the probability of rolling a 6 with a "fair" die, it is merely a hypothesis in a problem - in the same sense that a statement like "Ed has 6 apples" can be a hypothesis (a "given") in algebra problem. As far as the pure mathematics of probability theory goes, it is a special case of "measure theory". There is no distinct "frequentist measure theory" versus "Bayesian measure theory".

It is in applications of probability theory that there is a distinction between the Bayesian and frequentist approaches. To get mathematical solutions to real life problems, assumptions must be made. The frequentist and Bayesian approaches differ in how they choose to make assumptions. When it comes to applying probability theory to physics, there are many different interpretations of probability. In fact, there are many different interpretations of probability among people who call themselves Bayesians.

Textbooks on probability theory often try to be helpful by mixing the mathematics of probability with examples of its application. For example, the typical text on introductory statistics teaches the "frequentist" approach to applying probability theory. Modern textbooks about probability theory (as opposed being only about the topic of statistics) usually show both the frequentist and Bayesian approaches.

Thanks, Stephen. I looked up measure theory but it’s too advanced for me at the moment. Can you tell me in the meantime if I’ve understood part of what you said correctly. Pure maths, regards probability, is not based on any number of observed events (e.g, looking at thousands of spins of a roulette wheel) it’s purely theoretical. We ASSUME that with an unbiased roulette wheel and an unbiased croupier the ball has an equal chance of landing in any of the pockets and assign a probability accordingly (the P for each pocket is 1/37)?

“The frequentist and Bayesian approaches differ in how they choose to make assumptions.”

Could you give me some examples of how they might differ in their assumptions regards my example of red v non-red?

 

[Dale]

the objective reality would be that THERE IS ONLY ONE POSSIBLE OUTCOME,

In the Bayesian approach probability does not reflect objective reality. In the frequentist approach it represents the objective reality of the proportion in an infinite ensemble of experiments. So a frequentist cannot assign a probability for a single event.

 

[BWV]

“...the use of probability reflects our ignorance of the outcome - i.e. in the absence of any relevant information, any particular spot on the wheel is as likely as any other.”

I appreciate that assuming it is deterministic (as you say it is) we are still ignorant of what the outcome will be; but I don’t see how “any particular spot on the wheel is as likely as any other” if it is deterministic. I can see why from our subjective point of view it would seem like that, given our ignorance of the outcome, but the objective reality would be that THERE IS ONLY ONE POSSIBLE OUTCOME, i.e, the ball will land in a red pocket because that is what has been determined by antecedent events. Which is why I’m wondering if the actual probability for red is 1, and the actual probability for non-red is 0.

If what you say was true, then (outside of quantum mechanics) the odds for anything would be either zero or one and statistics would be of absolutely no use to anyone If I flip a coin, look at the outcome and ask you to guess heads or tails, then, yes, the probability depends on your ignorance. Just accept probability as a measure of ignorance about a system and stop overthinking it, like it represents some objective reality.

And ignore the bayesian / frequentist stuff - its a huge waste of time going down that rabbit hole.

 

[Dale]

And ignore the bayesian / frequentist stuff - its a huge waste of time going down that rabbit hole.

I disagree completely. The source of his confusion is trying to put a frequentist interpretation of probability where it cannot be applied. The OP needs to be aware that there is an alternative interpretation where it can be applied but in that interpretation probability is not objective.

Just accept probability as a measure of ignorance about a system and stop overthinking it, like it represents some objective reality.

Which is the distinction between the frequentist and Bayesian interpretations of probability, in a nutshell.

 

[BWV]

The Bayesian / Frequentist stuff does not apply to the OP, is is confused on a more basic level - confusing assumptions of probability with basic physics which are separate issues. If you can't apply frequentist definitions of probability to casino games, then where can you use it? (of course you can, this is where the whole theory developed) Its is not a B/F issue to state that probability involves our ignorance of certain attributes of a system, as outside of QM, this is where all probability comes from.

 

[Stephen Tashi]

Thanks, Stephen. I looked up measure theory but it’s too advanced for me at the moment.

One way to conceptualize measure theory is that it is an abstract treatment of the idea of "area" or "volume". A set (such as the possible outcomes of the toss of a fair coin, {H,T}) has certain subsets (such as {H},{T}) for which there is a function p() that assigns a "measure" to the subsets (such as p{H} = 1/2, p{T} = 1/2). No interpretation of what the measure means is given.

Can you tell me in the meantime if I’ve understood part of what you said correctly. Pure maths, regards probability, is not based on any number of observed events (e.g, looking at thousands of spins of a roulette wheel) it’s purely theoretical.

Yes. The pure mathematical treatment of probability does not specify any physical experiments for measuring probabilities.

We ASSUME that with an unbiased roulette wheel and an unbiased croupier the ball has an equal chance of landing in any of the pockets and assign a probability accordingly (the P for each pocket is 1/37)?

I'll say no.

Making that statement seems to assume that "unbiased" is concept from physics and that "chance of" has either a physical definition or a mathematical definition that is more fundamental than the phrase "probability of". If you define "unbiased roulette wheel" to mean a wheel where each outcome has an equal "chance of" happening and define "chance of" to mean "probability of" then you can make the above statement - however, the statement is then "true by definition"; it has no significant content other than being a vocabulary exercise.

We have to face the fact that a logically consistent concept of "probability" contradicts any attempts to make definite assertions that do not themselves involve the concept of "probability". Probability theory is circular in that respect. For example if the probability that a "fair" coin is tossed 100 times, we can say nothing definite about whether it lands heads 50 times or whether it lands heads between 40 and 60 times. We can only compute the probability that it lands heads 50 times and the probability it lands heads between 40 and 60 times. Probability theory only makes statements about probabilities.

The circular nature of probability theory is unsatisfactory from the point of view of confronting real life problems. However the circular nature of probability theory is essential from the point of view of doing mathematics. Attempts to make probability theory say something definite about things other than probabilities lead to metaphysical disputes. (e.g. Can an event with probability zero happen? If I toss a fair coin long enough, will I (definitely) eventually get 3 times as many heads as tails? etc.) Mathematical probability theory wisely does not touch such questions!

Could you give me some examples of how they might differ in their assumptions regards my example of red v non-red?

Let say you have a roulette wheel not know to be "fair".

A frequentist approach is to assume the probability of red is a some fixed, but unknown number $P_{red}$. We assume the wheel is fair, i.e. we assume $P_{red} = 18/37$. We compute the probability $p$ that in 3700 results from the wheel we would get between $1800 - T$ and $1800 + T$ reds. We choose $T$ so that $p$ is "small" (e.g. p = 0.05 is a typical choice). We observe 3700 outcomes of the wheel. If the number of reds is less than $1800-T$ or greater than $1800 + T$ we "reject" the hypothesis that the wheel is fair.

In the frequentist approach, the choice of $T$ is subjective. The statement that we "reject" the hypothesis is not the same as saying definitely "The wheel is not fair" or asserting that the probability the wheel is not fair is such-and-such. The frequentist approach is simply a procedure which has been found empirically to be useful in many fields of study.

A Bayesian approach would be to imagine the wheel in question was taken at random from a population of wheels and this results in some probability distribution for the number $P_{red}$. (In this example, you must swallow the notion of "a probability of a probability"). For example we might assume the $P_{red}$ is chosen from a uniform distribution of numbers over the interval [0.4, 0.7]. We observe 3700 outcomes of the wheel and observe $N_{red}$ reds. Then we compute the function that gives us the probability that $P_{red} = x$ given there were $N_{red}$ reds. (Imagine making a graph of this function.) Making a decision about whether the wheel is fair or not is done on the basis of that function and other assumed or known information that quantifies the expected gain or loss from making a correct or incorrect decision.

A fundamental distinction between the frequentist and Bayesian approaches is that the frequentist procedure involves computing the probability of the data given the hypothesis that the wheel is fair while the Bayesian procedure involves computing the probability that $P_{red}$ takes various values given the observed data. (There is a distinction between the conditional probabilities Pr(A given B) and Pr(B given A) ).

The frequentist and Bayesian procedures also differ in what subjective assumptions must be made. The Bayesian approach requires more elaborate assumptions.

People can offer various philosophical reasons for using the above approaches (e.g. that $P_{red}$ is an objective property of the physical wheel versus that $P_{red}$ is quantification of our ignorance about whether a red happens etc.). The mathematical procedures themselves do not specify particular philosophies or metaphysics. However, it's fair to say that metaphysical views about probabilities being a measure of information or ignorance can be used to justify Bayesian procedures.

 

[BWV]

A simpler Bayesian approach would be to say, that in the absence of other information, how the wheel is 'unfair' is unknown, so the prior for red would simply be
18/37, as the 'fair' odds would be the mean of a distribution of 'unfair' wheels. As the wheel is spun an arbitrary number of times and the 'unfairness' begins to emerge, then the Bayesian would update the distribution accordingly.

 

[Stephen Tashi]

A simpler Bayesian approach would be to say, that in the absence of other information, how the wheel is 'unfair' is unknown, so the prior for red would simply be
18/37

A Bayesian guess would be 18/37, but for a prior that can be updated, we'd need a distribution where other values are possible. (i.e. the prior distribution $Pr(P_{red} = 18/37) = 1$ can't be updated .)

 

[BWV]

A Bayesian guess would be 18/37, but for a prior that can be updated, we'd need a distribution where other values are possible. (i.e. the prior distribution $Pr(P_{red} = 18/37) = 1$ can't be updated .)

No, the P(red)=18/37 is the mean of the prior distribution of 'unfair' wheels

 

[Stephen Tashi]

No, the P(red)=18/37 is the mean of the prior distribution of 'unfair' wheels

Ok, but you'd have to specify what particular distribution is to be used. There is more than one probability distribution with mean 18/37.

 

[BWV]

Yes, but this whole thread has got off topic on the B/F issue as the OP was much more basic than that. As all non-QM physical systems are deterministic given (typically unknowable) perfect knowledge of its initial conditions then it follows all applications of probability to these systems is an acknowledgment of our ignorance of these conditions. The B/F stuff are methods in estimating these probabilities, but the OP, as I read it, is questioning the use of probability on deterministic systems

 

[Stephen Tashi]

but the OP, as I read it, is questioning the use of probability on deterministic systems

In particular the OP asks about the (true) values for probabilities of events in a deterministic system. Are they always either 0 or 1 ?

As all non-QM physical systems are deterministic given (typically unknowable) perfect knowledge of its initial conditions then it follows all applications of probability to these systems is an acknowledgment of our ignorance of these conditions.

To classify a system as "deterministic" versus "Quantum mechanical" involves assuming a model for it. Any method that acknowedges assuming a model acknowledges our ignorance of facts. On may assume a probability model for a phenomena without comitting to a particular philosophical viewpoint. about the interpretatipn of probability. One answer to the original post is that systems thought to be deterministic are often represented by models that use values of probabilities different than 0 or 1. The assumption of these values, and the assumption of the model itself is an acknowledgment and consequence of our ignorance. This does not say that the interpretation of the probabilities used within the model must involve a "measure of ignorance".

The general idea that probability is a measure of infomation or ignorance is not specific enough to use in interpreting probability models. To be specific, we have to say what event the ignorance concerns. For example, we can think of one fair coin tossed 100 times and say 1/2 is a measure of our ignorance that the coin will land heads on the 15th toss. Or we can think of an "ensemble" of fair coins that have each been tossed 100 times and each have definitely landed (either heads or tails) on the 15th toss. We can assign 1/2 as measure of our ignorance about which particular type of coin in the ensemble is the one we are dealing with.

The problem of determinism versus probability is but one aspect of the metaphysics of interpreting probability as involving an "actuaL" event that happens from a set of "possible events". In the mathematical treatment of probability as measure theory, there is no definition of "actual" versus "possible" events. There isn't even an axiom that says we can take samples of a random variable.

 

[FactChecker]

At the quantum level, there are believed to be outcomes that are intrinsically probabilistic -- not because we do not know enough to predict the outcome deterministically, but because the entities involved are truly probabilistic.

On the larger, non-quantum level, if we knew all that God knows and could do all the calculations, we could deterministically predict the outcome. Of coures, we are actually not in that situation and we have incomplete information and knowledge. In that case, probabilities should be thought of as the theory of guessing given incomplete information. A good example is the case where the outcome has already occured, like a prior toss of a coin that we can not see. Clearly the outcome is given and must be considered determined. Yet we have incomplete information and our guess of the outcome must be calculated as probabilistic.

 

[Stephen Tashi]

In that case, probabilities should be thought of as the theory of guessing given incomplete information.

In the mathematics section of the forum, we can discuss physical and metaphysical concepts, but (to me) it is important to distinguish mathematical ideas from ideas about physics and metaphysics.

In mathematical probability theory (meaning the measure theoretic approach) there are no definitions that deal rigorously with the physical concepts of "determinism" and the "actual" occurence of events versus "possible" events that might have occurred. So any questions of how probabilities are to be interpreted when applying mathematical probability theory to a certain real life problem fall outside the scope of mathematics and within the scope of whatever discipline studies the particular problem.

I agree that logic falls within the scope of mathematics. The original post does ask a question that could be considered purely a question of logic - it asks whether probabilities assigned to events in a deterministic process must be either 0 or 1. However, to answer that question, the subject matter to which we apply logic is not mathematical probability theory. The subject matter to which we apply logic involves interpreting concepts like "deterministic" and "actually occurred" in the context of situations in physics or metaphysics.

It is certainly true that people who apply probability theory to situations that are "deterministic" by some definition used in physics, justify their use of probabilities by interpreting the concept of probability as having to do with information or ignorance. The point I wish to emphasize is that this approach is not mandated or prohibited by mathematical probability theory. Mathematical probability theory says "No comment".

People whose background is in applied mathematics often find the rigorous presentation of probability theory repulsive because familiar concepts in applied math like random samples only appear in a ghostly form, such as conditional probability distributions. However, if we consider the many inconclusive debates that arise about the interpretation of probability, then the wisdom of Kolmogorov becomes apparent. He picked out the aspects of applied probability that can be treated rigorously and left the rest to Philosophy.

 

[cliffhanley203]

I'm thinking that maybe you don't understand probability.

This isn't a term that mathematicians use to describe themselves, to the best of my knowledge.

Are you questioning whether this probability is correct?

“I'm thinking that maybe you don't understand probability [re ‘I appreciate that, unless you were Laplace’s demon, you couldn’t know this in advance of the spin but I’m still wondering if, even though we would describe the probabilites as 18/37 and 19/37, they would actually be 1 and 0].”

If something literally cannot happen is the probability of it happening zero? And if something literally cannot not happen is the probability of it 1? For example (regards the former) is the probability of the ball on a roulette wheel landing in the pocket marked blue 38 zero (where there is no pocket marked blue 38)?

“This [frequentist] isn't a term that mathematicians use to describe themselves, to the best of my knowledge.”

I hadn’t heard it used either, until I posted this thread. But now I’m reading about it online.

“Are you questioning whether this probability [the P of rolling a 6 on a die is 1/6 etc] is correct?”

I wasn’t, until now. What I’m hearing now, on this thread, is that, for the frequentist, the probability of a single roll (or spin etc) “isn’t defined”.

 

[Stephen Tashi]

If something literally cannot happen is the probability of it happening zero? And if something literally cannot not happen is the probability of it 1? For example (regards the former) is the probability of the ball on a roulette wheel landing in the pocket marked blue 38 zero (where there is no pocket marked blue 38)?

Perhaps you mean to say "And if something literally cannot not happen is the probability of it not happening 1?"

Those are interesting questions, but they are not answered by the mathematical theory of probability. They deal with interpreting the mathematical theory in applying it to particular problems. So keep in mind that people answering those questions are giving interpretations of mathematics, not mathematical laws. (For example, the "frequentist" versus "Bayesian" positions are different interpretations of the same mathematics.)

 

[Dale]

However, if we consider the many inconclusive debates that arise about the interpretation of probability, then the wisdom of Kolmogorov becomes apparent. He picked out the aspects of applied probability that can be treated rigorously and left the rest to Philosophy.

I agree here. The axioms of probability can be used as a mathematical tool to model a system regardless of any philosophical interpretation of what probability means. All you need is the “minimal interpretation” where you map mathematical objects to observables of the system.

Both the Bayesians and frequentists use the same axioms, so in large measure the distinction is a matter of convenience. Use whichever version suits a problem best, including switching back and forth as convenient. Or use neither, simply use the axioms with the minimal interpretation. Although in the case of a single future roulette spin it is hard for me to see how to apply the axioms to anything other than subjective beliefs.

 

[stevendaryl]

In practice, it seems to me that there are three broad categories of how probability is used in science:

1. You have an example taken from a population of similar individuals. The individuals within the population differ in certain respects, such as size, or shape, or color, or whatever. You use probability to reflect the likelihood of the selected individual having various attributes.
2. You have a system that is stochastic: that is, the future state of the system is not uniquely determined by its past. Then you use probability to describe the likelihood of some future state given the current state.
3. You have a system that is assumed to follow some evolution law that contains parameters. You don't know the values of those parameters, so you use probabilities to describe the likelihood of various possibilities.

I guess I would say that only #2 makes sense from the point of view of objective probability, while #1 and #3 are compatible with a subjective notion of probability.

Of course, in practice, #2 could be an approximation, where the underlying source of probabilities is more like #1 or #3. For example, we model a coin flip as a stochastic process, with 50/50 chance of heads or tails, but at some level, it might actually be determined by details of the coin or our flipping process.

It seems to me that frequentist probability is really assuming something like #2, that there is a truly random process that only has probabilistic results. Frequentist probability is often applied to cases such as #1 and #3, but I think that it's justified by thinking of the selection process (for an individual out of a population, or for one parameter value out of a set of possible values) as a random process of type #2.

I wouldn't characterize Bayesian probability as being inherently about subjective probability, but instead I would say that Bayesian probability doesn't distinguish between subjective and objective probability. A common use of Bayesian reasoning is to use subjective probability on the parameters describing a system (#3), and then use the parameters to determine the stochastic probabilities (#2). For example, in analyzing a coin flip, a Bayesian might say:

• Assume the coin has a parameter $h$ that determines the stochastic probability of getting heads each flip. Assume that the parameter $h$ can take on any value from 0 to 1, with a flat probability distribution.

That mixes #2 and #3. Or if you think of different coins as having different values for the parameter $h$, then it might be mixing #1 and #2.

 

[cliffhanley203]

Perhaps you mean to say "And if something literally cannot not happen is the probability of it not happening 1?"

Those are interesting questions, but they are not answered by the mathematical theory of probability. They deal with interpreting the mathematical theory in applying it to particular problems. So keep in mind that people answering those questions are giving interpretations of mathematics, not mathematical laws. (For example, the "frequentist" versus "Bayesian" positions are different interpretations of the same mathematics.)

“Perhaps you mean to say "And if something literally cannot not happen is the probability of it not happening 1?"

Wouldn’t the P of it not happening be zero if it literally cannot not happen – which means it must happen)?

What I meant to say was if something literally cannot not happen is the probability of it happening 1.

Do you agree with the first part, i.e, that if something literally cannot happen the P of it happening is zero. Example; the ball landing in the pocket marked blue 38 zero when there is no such pocket?

 

[Dale]

You have to be careful with stuff like this. For example, if x is a standard normal random variable, then whatever value you obtain from sampling x, the probability of getting exactly that value was 0. So p=0 doesn't necessarily mean that it cannot happen.

 

[Stephen Tashi]

Wouldn’t the P of it not happening be zero if it literally cannot not happen – which means it must happen)?

You have a fundamental misunderstanding. "Probability" is a number associated with a set. If E is a set, P(E compliment) = 1 - P(E). If P(E) = 1 then P(E compliment) = 0.

The mathematical probability of an event has no definite connection with the non-mathematical concepts of "happening" or "not happening".Having said that, the usual way to formulate a probability model model for a real life problem is to assign higher probabilities to events that "happen" more frequently. The usual way to interpret the set E compliment is that it represents the event that the things in set E did not happen.

 

[cliffhanley203]

You have to be careful with stuff like this. For example, if x is a standard normal random variable, then whatever value you obtain from sampling x, the probability of getting exactly that value was 0. So p=0 doesn't necessarily mean that it cannot happen.

Q. Is the following random variable a ‘standard normal random variable’;

X = {0,1,3...all the way to 36) [with reference to a European roulette wheel] [and the numbers in the brackets being the sample space]?

Q. Re ‘...whatever value you obtain from sampling x...’ ; do you mean spinning the wheel several times and collecting some data? Or does 'sampling' mean putting the possible outcomes in brackets as I've done above?

P.S. Thanks for your patience.

 

[Dale]

Is the following random variable a ‘standard normal random variable’;

No, a standard normal random variable has a normal (bell shaped curve) distribution with mean 0 and standard deviation 1. It is a continuous distribution, not a discrete one like a roulette wheel.

‘...whatever value you obtain from sampling x...’ ; do you mean spinning the wheel several times and collecting some data?

Yes

 

[cliffhanley203]

“You have a fundamental misunderstanding. "Probability" is a number associated with a set. If E is a set, P(E compliment) = 1 - P(E). If P(E) = 1 then P(E compliment) = 0.”

Where is my fundamental misunderstanding? Complement means, in the above example, ‘not E’, or, not the event/set in question, or, to quote Mathsisfun, “...the Complement of an event is all the other outcomes (not the ones we want). "

Just to check that I have got it, I’ll apply it to my example of the P of red on a European roulette table where red is determined (in a determinstic world).

The set, call it R (for red), is the all of the red numbers on the wheel. So R = {1,3,5,7,9, etc}

The complement of R is all the numbers that are not R (that is black and green), or, R’. So, R’ = {0,2,4,6,8,10,etc}.

And, to lay it out as you have above;

If R is a set, P(R complement) = 1 – P(R). If P(R) = 1 then

P (R complement) = 0.

So, using the classical interpretation, in a indeterminstic world;

P(R) = 18/37. P(R complement) = 1-18/37 = 19/37.

And, finally, using the classical interpretation in a deterministic world;

P(R)= 1. P (R complement) + 1-1 = 0.

“The mathematical probability of an event has no definite connection with the non-mathematical concepts of "happening" or "not happening".”

Q. What do you mean by no “definite” connection?

Q. What connection does it have with those concepts?

“Having said that, the usual way to formulate a probability model for a real life problem is to assign higher probabilities to events that "happen" more frequently. The usual way to interpret the set E compliment is that it represents the event that the things in set E did not happen.”

Yes.

 

[FactChecker]

“The mathematical probability of an event has no definite connection with the non-mathematical concepts of "happening" or "not happening".”

Q. What do you mean by no “definite” connection?

Q. What connection does it have with those concepts?

In this context, it is worth thinking about @Dale 's post #31. Suppose that a number has been selected from a continuous distribution. Then something has happened. Call that selected number r. Then r has "happened" but its probability is 0. That exact number, r, will never happen again.

 

[Dale]

In this context, it is worth thinking about @Dale 's post #31. Suppose that a number has been selected from a continuous distribution. Then something has happened. Call that selected number r. Then r has "happened" but its probability is 0. That exect number, r, will never happen again.

Also, the complement of r has probability 1.

 

[Stephen Tashi]

Where is my fundamental misunderstanding?

This is what you wrote:

Wouldn’t the P of it not happening be zero if it literally cannot not happen

Yes, your example corrects this statement.

Q. What do you mean by no “definite” connection?

Assigning an event a probability does not permit us to make statements about the event happening or not happening unless those statements also speak of probabilities. If we say Pr( "The probability the coin lands heads") = 1/2, this does not imply the statement "The coin will land heads 50 out of 100 times".

As another example, there is deductive principle in logic known as modus ponens that says:

Given:
If A then B
A is true
Conclude:
B is true

There is no principle in logic that allows the argument:

Given:
If A then B
A has probability 1 of being true
Conclude:
B is true

In applying probability to specific problems, people may assume the latter argument to be valid, but the latter argument is not a general principle of logic. It may be valid in specific situations and specific interpretations of probability models.

Q. What connection does it have with those concepts?

Assigning probabilities to whether events happen allows us to make deductions about the probabilities of other events happening.

 

[cliffhanley203]

“Yes, your example corrects this statement.”

So, just to be clear; using the classical interpretation in a deterministic world where it’s been determined that the ball will land in a red pocket the P of red is;

P(r) = 1.

P(r complement) = 1-1=0 ?

 

[Stephen Tashi]

So, just to be clear; using the classical interpretation in a deterministic world where it’s been determined that the ball will land in a red pocket the P of red is;

P(r) = 1.

Yes, that is usual interpretation for the probability of an event that definitely must happen.

P(r complement) = 1-1=0 ?

Yes.

 

[cliffhanley203]

“Assigning an event a probability does not permit us to make statements about the event happening or not happening unless those statements also speak of probabilities.”

Even if the P is 1? Or if the P is O? I was under the impression that if the P is 1 it must happen, and if the P is 0 it can’t happen; because if the P is said to be 1 and it doesn’t happen then the assertion that the P was 1 was incorrect; and if the P is said to be zero and it does happen then the assertion that the P was zero was incorrect.

“If we say Pr( "The probability the coin lands heads") = 1/2, this does not imply the statement "The coin will land heads 50 out of 100 times".

I get that. In the short run we could see 60 heads, 40 tails, or 80 heads, 20 tails, or even 100 heads, 0 tails. But the more we toss the coin the closer we will get to the expected value. Which is demonstrated by the profitability of roulette for casinos. I even hear mathematicians say that if you tossed the coin an infinite number of times you would get exactly 50% heads, 50% tails. Or if you spun the roulette wheel an infinite number of times you would get exactly 18/37 red, 19/37 non-red. Although that seems to me to be impossible to know so I don’t know how they would justify that assertion.

 

[stevendaryl]

Even if the P is 1? Or if the P is O? I was under the impression that if the P is 1 it must happen, and if the P is 0 it can’t happen; because if the P is said to be 1 and it doesn’t happen then the assertion that the P was 1 was incorrect; and if the P is said to be zero and it does happen then the assertion that the P was zero was incorrect.

If the probability is 1, it means you should definitely bet on it happening, but it doesn't mean that it must happen. Similarly, probability 0 means that it almost surely will not happen, but not that it is impossible to happen.

Think about flipping a coin repeatedly. It's not impossible to get all "heads", but it's vanishingly unlikely.

 

[cliffhanley203]

In this context, it is worth thinking about @Dale 's post #31. Suppose that a number has been selected from a continuous distribution. Then something has happened. Call that selected number r. Then r has "happened" but its probability is 0. That exact number, r, will never happen again.

But it's probability wasn't zero prior to it happening, that would be impossible. If, in a deterministic world, it is determined that red will appear, the P of non-red appearing is (using the classical interpretation) zero. In this case non-red can't happen (if it does then it wasn't determined that red would appear).

"Then r has "happened" but its probability is 0. That exact number, r, will never happen again"

Shouldn't that be; Then r has happened and now that it's happened and will never happen again it's probability is 0?

 

[cliffhanley203]

“No, a standard normal random variable has a normal (bell shaped curve) distribution with mean 0 and standard deviation”

Thanks, Dale, but that’s a bit advanced for me at the moment.

“So p=0 doesn't necessarily mean that it cannot happen.”

Is this an instance of the P of something being zero prior to it happening, and then it happening? If so, wouldn’t it's happening contradict the earlier assertion that it’s P was zero?

 

[cliffhanley203]

If the probability is 1, it means you should definitely bet on it happening, but it doesn't mean that it must happen. Similarly, probability 0 means that it almost surely will not happen, but not that it is impossible to happen.

Think about flipping a coin repeatedly. It's not impossible to get all "heads", but it's vanishingly unlikely.

Thanks, Steven. But I'm talking about a deterministic world where it's been determined that red will win. If, in a deterministic world, it was determined that the next toss of a coin would come up heads then the P for heads is 1, the P for heads is 0, yes? And heads must happen; tails can't happen (anything else would contradict the fact that it was determined to be that way).

 

[StoneTemplePython]

“I even hear mathematicians say that if you tossed the coin an infinite number of times you would get exactly 50% heads, 50% tails.

Let's hope not. It's well known that equalizing (i.e. having exactly $\text{count} = \text{number of heads} - \text{number of tails}= 0$) will happen with probability 1, and hence infinitely many times over these trials. However the expected amount of time until equalization (renewal) is $\infty$ which immediately tells you that the probability of being equalized at any time tends to zero. Infinity is a delicate business.

So suppose a different interpretation of the statement is needed -- they may have been referring to averaging many different trial's counts -- and that average tends to the a count of zero. Or they may have been averaging (read cesaro mean) the score of the trials you are quoting, then applying strong law of large numbers (I suspect that this is what is being done, though the statement has a different interpretations as a symmetric random walk.)

- - - -
As for your other questions -- impossible events have probability zero. But not all probability zero events are impossible. There are a lot of subtleties related to infinity that keep coming up in these questions, over and over. Bottom line -- wielding infinity properly takes a lot of work and care.

- - - -
Btw, even if you discover world is 'truly deterministic' I don't think that is going to change how your car insurance, life insurance, etc. is priced.

 

[stevendaryl]

Thanks, Steven. But I'm talking about a deterministic world where it's been determined that red will win. If, in a deterministic world, it was determined that the next toss of a coin would come up heads then the P for heads is 1, the P for heads is 0, yes? And heads must happen; tails can't happen (anything else would contradict the fact that it was determined to be that way).

Sorry, I missed the context. It seems to me that there is no point in talking about probability if everything has probability 0 or 1. But yes, if things are deterministic, then an event is either definitely going to happen, with probability 1, or is definitely not going to happen, so it has probability 0.

On the other hand, even in a deterministic universe, we could probability to quantify our lack of information about the current state of the world, in which case we would introduce probabilities that are neither 0 nor 1, and we would also introduce the possibility of something being given a subjective probability of 0 even though it wasn't actually impossible.

 

[Stephen Tashi]

Even if the P is 1? Or if the P is O?

You are correct that a common way to apply probability to real life situations is to assign a probability of 1 to events that must happen and a probability of zero to events that cannot happen.

(As I keep saying, the mathematics of probability theory does not deal with the topic of events "actually happening" or "actually not happening". That subject is a matter of interpreting the matematical theory when applying it to specific situations. By analogy, the mathematical theory of trigonometry does not make specific claims about ladders, distances between cities, heights of trees etc. Those topics involve interpreting the mathematical theory when applying it to specific situations.)

The difficulties in interpreting probabilities of 0 or 1 arise in the applying probability theory to outcomes from an infinite set of values. Your roulette examples have only a finite number of outcomes, so interpreting probabilites of 0 or 1 in those examples is not controversial.

 

[Dale]

So, just to be clear; using the classical interpretation in a deterministic world where it’s been determined that the ball will land in a red pocket the P of red is;

The classical interpretation of P is not defined in such a case. I feel like this is getting annoyingly repetitive.

The classical interpretation of probability is the ratio of successes (red) over the course of an infinite number of trials. It simply is not defined for a sample space consisting of a single unique event. There is no classical interpretation of probability which can apply to this case.

If you wish to extend this to a scenario where the classical interpretation of probability does apply then you will need to extend it beyond a single determined roulette spin to an infinite set of determined spins. Then the question is which infinite set of spins are you considering? Are you considering an infinite set where every spin is determined to be red, or are you considering an infinite set where every spin is determined but what it is determined to be varies. The classical probability does not depend on the determinism, only the long run frequency in the infinite set under consideration.

I was under the impression that if the P is 1 it must happen, and if the P is 0 it can’t happen;

I gave an explicit counterexample where the thing that happened had a probability of 0 and the thing that had a probability of 1 did not happen.

But it's probability wasn't zero prior to it happening, that would be impossible.

Yes, it was. That is the whole point.

Thanks, Dale, but that’s a bit advanced for me at the moment.

Then maybe you should focus on the basic mechanics of using probability before you get too deep into the philosophy. I can’t help but feel that you are tying yourself in mental knots here.

https://en.m.wikipedia.org/wiki/Normal_distribution

Is this an instance of the P of something being zero prior to it happening, and then it happening?

Yes.

If so, wouldn’t it's happening contradict the earlier assertion that it’s P was zero?

No. In the frequentist case probability is defined as the ratio over an infinite number of trials, and that ratio would still converge to 0. In the Bayesian case it represents our prior subjective belief. Either way the actual result does not contradict the assigned probability.

 

[PeroK]

Looking at many comments on this thread it's interesting how deeply ingrained the idea of determinism is in (non QM) physics.

I don't agree with this at all. It seems to ignore the theoretical unknowability of all the information associated with a complex dynamic system, such as weather or human behaviour.

Those who think physics can predict a coin toss seem to assume that they are able to wait until the coin has been tossed. But, you only have to postulate a future coin toss with unknowns - where and precisely when the toss is to take place, the choice of coin, and the choice of who tosses the coin - to rule out the physics solution.

And, if you believe you could use physics to predict all future human behaviour, then how do you gather the information without changing the people involved?

It's the same with weather systems. Whatever equipment you put into the atmosphere to measure things become part of the system and need themselves to be measured. Plus, the weather is fundamentally affected by the Sun. So, you'll need probes throughout the Sun to predict the weather on Earth.

And, of course, you cannot have probes everywhere. as then the Sun and the Earth word be literally nothing but those probes.

The absolute determinism postulated in this thread is a theoretical impossibility. Probability remains at the heart of physical phenomena, even without the atomic and subatomic uncertainties modeled by QM.