B How might determinism affect probability?

[cliffhanley203]

“It seems to me that there is no point in talking about probability if everything has probability 0 or 1.”

I think I know what you mean but I’ll ask why anyway?

“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.”

That’s what I thought was the case but I’m not so sure anymore given the various interpretations (classical; frequentist; bayesian).

“On the other hand, even in a deterministic universe, we could [use] 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...”

But in doing so we would be, technically, incorrect to say that an event had a P of neither 0 nor 1 when it actually does have a P of either 0 or 1 (objectively speaking).

“...and we would also introduce the possibility of something being given a subjective probability of 0 even though it wasn't actually impossible.”

Yes, a subjective P (but the objective P would be fixed as 0 or 1).

 

[cliffhanley203]

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.

So there are basically two types of probability. The probability of pure maths (theoretical only, like a Euclidean line – infinite in length in both directions). And there’s applied probability (casinos realising that they can’t lose provided they get the punters money on the table for long enough to see the expected value of their house edge)?

“(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.)”

But trigonometry works when applied to ladders, distances etc, yes? As does probability applied to roulette, yes?

 

[Stephen Tashi]

Can you give a simple example of an even with a P of zero that is possible?

Imagine a continuous roulette wheel where instead of pockets, there is smooth track of length 37. If we want to say there is an equal probability of the center of the ball stopping anywhere, we can assign a probability of 1/37 to it landing between 6 and 7, a probability of 1/74 of it landing between 6 and 6.5 , etc. However, we must assign the probability of zero to the event it lands exactly on a given point such as 6.5. Yet (in theory) it does land exactly at some point.

 

[Stephen Tashi]

But trigonometry works when applied to ladders, distances etc, yes? As does probability applied to roulette, yes?

That wouldn't show that the facts of life in roulette determine the facts of life about probability. The fact that theory A empirically works when interpreted as real world situation B does not imply that the two are the same thing - or that situation B is the only way to interpret theory A.

There is an overwhelming desire (even among contributors to this thread who understand the Kolmogorov formulation of probability theory) to cast probability theory as a theory covering topics such as information or guessing or situations where events actually happen or don't or happen, or must happen in a infinite number of trials etc. However, none of these topics are covered by the mathematical theory of probability. All these topics involve interpreting probability in the context of some real or imagined situation.

 

[FactChecker]

There is an overwhelming desire (even among contributors to this thread who understand the Kolmogorov formulation of probability theory) to cast probability theory as a theory covering topics such as information or guessing or situations where events actually happen or don't or happen, or must happen in a infinite number of trials etc. However, none of these topics are covered by the mathematical theory of probability.

If you mean that probability theory does not concern itself with several of these issues and is agnostic, then I agree. But if you are saying that probability theory does not apply in these situations, then I'm afraid that I have to disagree.

 

[PeroK]

Imagine a continuous roulette wheel where instead of pockets, there is smooth track of length 37. If we want to say there is an equal probability of the center of the ball stopping anywhere, we can assign a probability of 1/37 to it landing between 6 and 7, a probability of 1/74 of it landing between 6 and 6.5 , etc. However, we must assign the probability of zero to the event it lands exactly on a given point such as 6.5. Yet (in theory) it does land exactly at some point.

Or, in theory, the ball is never truly at rest. Or, in theory, beyond a certain a certain accuracy the centre of the ball is not physically well-defined.

In a way, QM comes to the rescue. We could reduce the ball to a single elementary particle: it's not possible to know where the particle is without measuring it. And, any measurement has its margin of error.

And, the argument that at some time $t$ the particle was definitely at some physical location $x$ also fails.

If you don't like the idea that everything that is happening has probability zero, then in a way QM comes to the rescue.

In general, you cannot pretend that classical physics ultimately extends to point like particles and point like positions. With these arguments you have to take into account a non-classical, sub-atomic theory. Namely, QM.

 

[Stephen Tashi]

If you mean that probability theory does not concern itself with several of these issues and is agnostic, then I agree.

That's what I mean.

 

[StoneTemplePython]

“Let's hope not[that some mathematicians say that if you tossed the coin an infinite number of times you would get exactly 50% heads, 50% tails.].”

Thanks, I will challenge the next one that tells me that.

just ask the person to clarify what they mean -- it can be interpreted a couple ways (or more).

“It's well known that equalizing (i.e. having exactly count = number of heads − 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 ∞∞ which immediately tells you that the probability of being equalized at any time tends to zero. Infinity is a delicate business.”

I Googled equalizing but I wasn’t sure from the results which one applied to what you said. What do you mean here by equalzing?...

I defined equalizing as having exactly the number of heads tosses equal to the number of tails. It isn't a technical term per se (renewal would be) so you can call having an equal count something else if you want.

Yes, Dale (kindly, and patiently, explained this, but his example was too advanced for me – I’m, as you will probably have gathered) a novice). Can you give a simple example of an even with a P of zero that is possible?

@stevendaryl 's coin tossing example seems fine here to me. If you want to unravel this a bit -- suppose you toss a coin with probability $p$ of heads, with $p \in (0,1)$. Now toss the coin $n$ times. What's the probability that you see no tails ? That is every toss is a "heads" -- $p^n$. Now let $n \to \infty$ and you see the probability of all heads tends to zero.

I think a lot of people on this forum like Morin's Probability for Enthusiastic Beginner -- maybe worth looking into getting this -- it's relatively light on math background and has solutions to problems.

 

[Dale]

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.

@cliffhanley203 if you are interested in the classical interpretation of probability for your example then you must answer the above question

 

[FactChecker]

Two cases where, IMO, determinism fails are at opposite extremes: very small (quantum mechanics) and very large (entropy). In QM, Bell's theorem rules out the "hidden variable" theory. And entropy, by its very nature, deals with the probabilities of configurations of a very large number of items. I don't think that there is any way to use "deterministic" analysis to explain either one.