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What is the difference between 0 Probability and Impossibility?

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I might be wrong, but the way I see it is that getting a 7 on a dice is impossible - in that the outcome isn't an element/subset of the sample space.

But picking a variable from a continuous distribution at a fixed value (picking a height at exactly 180cm) is possible, yet it is a finite number of possibilities out of infinite others - in that it has 0 probability but nonzero possibility.

(If you keep picking heights randomly, it may take you infinite trials to reach 180cm, but it is possible to do so. If you roll a dice infinite times, no matter how many trials you do, 7 will never be an outcome.)

E2A: There's quite the discussion in the replies, I encourage anyone coming across this thread to read through

What I grasped is that our mathematical machinery draws no distinction between impossible and improbable events. Both of these 0 probability events can be concluded to have 0 probability in exactly the same ways.

The line between them literally isn't there and we're the ones going around drawing absurd seeming shapes to keep things consistent with the world we're familiar with.
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One means a set is empty.

The other means the set has measure zero under some probability measure. Such sets can be far from empty (e.g Cantor set)
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The short (and loose and nonrigorous) version is that impossible events literally can never happen; 0 probability events can happen but on average never do.

A simple example: consider flipping a coin infinitely many times.

Can it always land on heads? Sure. This is obvious.

The probability after n flips of having n heads is (1/2)^n. Taking n to ∞ then, the probability is zero, but it can happen.

Now what about all flips being bananas? Obviously never can happen (the statement itself is nonsense). The probability is still zero but because it can never happen.
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Maybe one should keep in mind that 0 probability is a well defined phrase in mathematics, whereas impossibility is a word that we use in regular speech.

In this thread people offer their adhoc definitions of impossibility, but I have never seen a definition in a text book (in the context of probability theory), nor do I think this concept would serve any real purpose in probability theory.

Of course this is not to say that the question is not interesting to think about.
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Saw this thread an immediately thought of the Sleeps drama from a few years ago lol
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There are no practical examples of possible events with 0 probability.

In theory, we can think about height as being any real number, of which there are (uncountably) infinitely many. Then the probability of any individual value, say 1.80cm, is zero, while ranges (say 1.80-1.80001) will have a nonzero probability.

Any real-life obeservation is going to be from a finite set of values, though. I.e. you observing a height of 181,3cm will have a nonzero probability for instance. Even with computers you'll be using floating point numbers that only have a predefined number of distinct values.
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In finite probability spaces, like rolling a die, "impossible" and "probability zero" are fundamentally the same. You can remove every event which has probability zero, or add however many probability-zero events you like, and the probability distribution is essentially unchanged. No trial you run will select an event which has probability zero of occurring. This is also true of some countable probability spaces, but that's not important here.

Say you pick a point randomly on a 1 meter by 1 meter square.

You can't do probability on this like you would for, say, rolling a six-sided die. There are infintely many (in fact uncountably many) events (points on the square which could be selected), so if we assume that each point is equally likely to occur, we would be adding up the same number infinitely many times... which obviously would give you either infinity or 0. But the probability of landing on any of the points is one, and in the land of finite probability spaces, summing up the probabilities of every point needs to give you 1.

So, if we try to do probability the same way on infinite and finite probability spaces, you get a contradiction.

Instead, we go another way: suppose you split the square into a grid of tiny squares, each of the same area, and suppose that the probability of picking a point in each of the tiny squares is equal. This is now a finite probability problem, and it works exactly how you would think it works. Landing in a particular square is just like rolling a particular number on a many-sided die.

Now suppose that this were true for *every possible grid of tiny squares*. Then you can start to do stuff like take limits, and in the end you will find that the probability of landing in any set is equal to the area of that set.

But... points don't take up any area, so the chance of randomly picking any *particular* point is zero, even though every point could possibly be picked. However, anything outside the square is simply impossible: if your random process always picks points from a 1x1 meter square, you will never get "banana".

(This is something a lot of people get wrong when philosophizing: the fact that there are infinitely many possibilities does *not* imply that literally anything is possible)

This unintuitive result was just the consequence of a completely reasonable, intuitive fact about dividing the square into tinier and tinier squares. It's just how the math works out if you want things to make sense no matter how tiny the squares are.

The notion of probability density functions helps us reason about continuous probability (where "probability zero but not impossible" can show up) in a way that doesn't have problems with "probability zero". The probability density function p(x) of a point x is the limit of the ratios P(landing distance less than r from point x)/(area of the circle with radius r) as r goes to 0. What's more, we can "sum up" a probability density function by taking its integral, which will always give us 1, showing a consistency with finite probability. In the example above this ratio is always one, so the pdf says something that we were trying to express from the start: every point is "equally likely" in some sense!
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A person will never be -20cm tall. That is outside the range of all possible: it is impossible. A person will never be exactly 180cm tall but it is a possible value. The neighbourhood of 180cm jointly has non-zero probability. The neighbourhood of -20cm is full of physically impossible values and thus jointly has 0 probability.

That is the difference between 0 probability (plausible value but this exact value will never occur) and impossible.

More formally, you might want to google around in analysis materials: The keyword „almost“ shows up a lot. For example, consider a function which is 1 everywhere  on the unit interval but 0 for x=0.5. this function is 1 almost everywhere. That’s math talk for: a infinite set of x values is associated with function value 1 and only a finite set of values is different (here x=0.5). Whenever you have a finite set of outcomes in an infinite sample space (eg the real numbers in the unit interval), the probability of the finite set occurring is 0: if we sample a random real number between 0 and 1 it will practically never be 0.5 and hence f(x) will be 1 „almost always“ but it is technically possible to get x=0.5. this is the equivalent to the 180cm on the example above. The -20cm would be like asking what the function value would be for x=2 even though we only defined the function between 0 and 1: it is outside our sample space.
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I guess to understand it well you have to learn about Measure Theory. You can measure some objects which have a measure of zero and still they exist. A point has no length, a line has no area etc...
A probability law can be understand as a form of measure on events, you can have events with probability 0, and still they can happen. If you randomly choose *uniformly* a real number between 0 and 1, only range of numbers, intervals have a probability equal to their length. A real number, a singleton has a probability of 0.
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Ha ha literally nothing

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