Hello,

In probability math,because of math's nature that is merely quantitative and not a qualitative, for any case,it give you just a number; so, I think for every cases, there should be a boundary probability number that is " meaningfulness " just for that specified case and out of that boundary is not meaningful and that is just a meaningless number.
For example,when you read the word;" text " ,the math probability of misunderstanding of that word is not zero but in real world is zero!

Thanks,

...what do you mean by " text "?

Forget the real world.

https://en.wikipedia.org/wiki/Information_theory

https://en.wikipedia.org/wiki/Entrop...rmation_theory)


PairTheBoard

Not sure I fully understand what you are saying, but perhaps you might find these interesting:

- With regard to "zero" probability:

https://en.m.wikipedia.org/wiki/Sunrise_problem
https://en.m.wikipedia.org/wiki/Additive_smoothing

- Bridging the quantitative and the qualitative in a different way:

https://en.m.wikipedia.org/wiki/Fuzzy_logic
https://en.m.wikipedia.org/wiki/Rough_set

Juk

very thanks for your reference; I should read them.

just the word ; " text " as an example.

Look up "almost surely" and "almost never". They are synonymous with probability 1 and 0.

Probability one does not mean must happen, and probability 0 does not mean can't happen.

I think you might also be describing the concept of "significance" which is quantifiable.

1 means 100%

Mathematically and literally it does. And that is absolutely not the same thing as "must happen" under probability theory. Infinity never gets here. The correct term is "almost surely".

With thanks for your answers; I meant that ; sometimes we get a probability ratio of for example of : 1/ 5 ∧ 23 from a calculation , but,in fact,a number of; 1/ 5 ∧ 8 for that is perfectly the absolute answer.

You can take all the beans you want that happen with probability 0 but they'll never add up to a hill of beans. Truth is, such beans never really happen because you never get exact results in reality. Probability spaces are mathematical models and you can imagine beans of probability 0 happening in the model but that's just a curio of the model, a figment of your imagination.


PairTheBoard

If you want to expand your view from thinking of locations as "points" you might study distribution theory. Instead of "points" you have test functions which are like smeared out locations. Distributions are generalized functions which are evaluated at these generalized "locations" by integrating them against the test functions.

https://en.wikipedia.org/wiki/Distribution_(mathematics)


PairTheBoard

As you know that we faced to a few cases(events),in our real life daily,that their occurrences are inevitable
but their math probabilities are still get you numbers that show uncertains!!

You have a misconception on this. Zero probability events happen all the time in real life. That's why it's defined as "almost never" in probability theory, rather than as impossible. Conversely, impossible events can have a non-zero probability, but for other reasons can't ever happen. Examples of both have been discussed before in the Probability forum on here.

Think about real numbers, which have infinite values in between any two numbers. Dividing by infinity = 0, but real events happen in that space all the time. Randomly choosing a specific real value has a probability of 0, yet it still happens in many many ways in real life.

When things happen there was some probability after all. Is rationalization after the fact allowed?

But basically yes. There may be zero probability for life to develop. Say the universe is eternal.

But can probability 0 happen twice?

Anytime there are an infinite number of outcomes, the probability is literally 0. But one of those outcomes actually happens every time. It isn't "some probability" or a rationalization at all. It's just how we define probability.

And nothing prevents the same one from happening twice, even though that is also probability 0.

You can talk about doing that for the sake of argument in the context of a mathematical model but good luck actually doing it.

Of course this is not true. Just consider P(k) = (1/2)^k for k=1,2,3,...

PairTheBoard

I'm not sure what I'm not explaining well, but it happens all the time that an event from an infiinite sample space happens, and a prediction for that event would have been probability = 0. This is well discussed in the literature, I was just pointing it out when it came up ITT.

Obviously we can say it's a quirk of probability theory, and also argue that we can't measure anything to infinite granularity etc, and with real world observable chunks, there is a >0 probability. But that isn't how it's defined.

There may be no real world events with a probability of 0. Let's throw in Planck time. Let's assume the universe isn't infinite (which it may be). Let's also assume we don't have an infinite number of universes too choose from.

Here's a real example.

Take a household electrical AC current which is generated according to a sine wave of voltage over time. The voltage continuously varies from -N to +N volts every cycle. Let's say the peaks are 170V giving us our RMS value of around 120V in the wall outlet. But during the sine wave cycle, the voltage takes on every possible value between the peaks.

So what is the probability that at a given moment in time, the voltage is exactly 120V? Since we have infinite values in the sine curve, this probability is x/infinity = 0.

Yet every single cycle, there is a moment in time where the voltage is exactly 120V. The probability 0 event happens over and over.

How do you know it varies absolutely continuously? Maybe it never is exactly 120V?

Are you just trolling now? Pick any value. It's a sine wave. If your argument is that probability theory in practice is only precise with physical perfection, that's true. The point is about theory, i.e. how things work and can be predicted in an idealized system. It's also true that physical infinities don't exist as far as we know (or can know), so that blows up everything then.

Edit: also, in an infinite set, you can remove any number of values and the set is still infinite. The probability is still 0 if some values don't occur in our sine wave (even though it's unlikely, a voltage increase physically can't happen instantaneously, it has to rise or fall. Unless of course you want to now argue at the electron level that it must change by discrete electrons. Do you know for sure that electrons are discrete even? ) We can do this all day.

Yes, right now I'm talking about reality, not idealized systems. Looks either one of us can be right in that case.

If you believe infinite sample spaces exist then in reality, probability 0 events happen all the time in many contexts. If you believe infinite outcomes can't really exist, then a lot of probability theory is problematic. I would argue that the example I gave above really exists in the physical world, and so do many like it. Anything measured on a continuous spectrum has infinite values even if our measuring and observing ability is granular and limited.