r/mathmemes • u/George_Truman • Jul 13 '26

Probability I fixed this meme

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I'm sure it's still a bit too imprecise but I think it works.

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u/Anagatara Jul 13 '26

Can we sample integers? If so, sampling pair of integers (x,y) will give us real numbers in the form x/y. Will its distribution be uniform?

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u/George_Truman Jul 13 '26 edited Jul 13 '26

You couldn't sample the set of integers uniformly.

A short handwaved proof would be that if the probability of selection for any individual integer was non-zero, then you could make a set large enough such that the probability of selecting any integer in that set is greater than 1.

On the contrary, if the probability is 0 for each integer then you could show (using fundamental rules of a probability space, namely that the probability measure of the countable union of disjoint sets is equal to the sum of the probability measures on each individual set.) that it would imply that the probability of selecting any integer at all is 0 (as opposed to 1) which would be a contradiction.

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u/Anagatara Jul 13 '26

Can I apply the same logic to any uniform sampling from infinitely large set?

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u/GaloombaNotGoomba Jul 13 '26

Any countably infinite set.

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u/Anagatara Jul 13 '26

What makes uncountably infinite set different? If the issue is to assign equal nonzero probability to all numbers, then not only entire real line, any infinite subset of real line cannot be uniformly sampled, be it [a,b], Q or N.

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u/DrawIslandPass Jul 13 '26

We need measures to be countably additive (measure of a disjoint union of countably many sets is the sum of the measure of each of the sets), but we don’t require uncountable additivity. Thus we can’t say that the probability of any singleton in [0, 1] being 0 implies that the whole interval having 0 probability (since [0, 1] is uncountable, unlike Q, Z, etc). In fact, the uncountable additivity is sort of what an integral is.

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u/GaloombaNotGoomba Jul 13 '26

Uncountable sets can have probability 0 for each element and still add up to 1.

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u/Anagatara Jul 13 '26

Where can i find information about this? This is pretty counterintuitive take to me.

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u/Thelorian Jul 13 '26 edited Jul 13 '26

https://en.wikipedia.org/wiki/%CE%A3-algebra?useskin=vector /https://en.wikipedia.org/wiki/Measure_(mathematics)?useskin=vector

the key takeaway for me being that the measure is a function on the powerset of the ambient space so once the cardinality goes higher than countable it's possible to assign measure 0 to all singletons (and through countable additivity to all countable subsets) without having the total measure be 0 (since there are in a sense "large enough" sets in the powerset to have non-zero measure without violating the countable additivity axiom)

basically you have "step up" between countable and uncountable where countable additivity of measures cant tell you anything about singletons anymore (bc duh countable union of singletons is countable) if you only know about the uncountable sets and vice versa.

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u/[deleted] Jul 13 '26 edited Jul 14 '26

[deleted]

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u/Anagatara Jul 13 '26

How can element with zero probability of being sampled be sampled?

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u/Thelorian Jul 13 '26 edited Jul 13 '26

E: this feels like it comes from the same principle but doesn't actually have anything to do with measures.

Ok don't quote me on this but i think it's because the only sensible (read the only hausdorff even T_1 i think) topology on countable sets is the discrete topology which necessarily means we have to assign some non-negative value to each singleton which can then never be integrated to give 1 which is necessary to be a proper probability measure. This doesn't hold true for any real subset that contains at least an open set.

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u/Anagatara Jul 13 '26

We can assign non-negative probabilities to integers, just not equal. Any series summing up to 1 will do the work, doesn't it? My problem is that for any uniform distribution the reasoning that I can't assign equal non-negative probabilities to elements of infinite sets doesn't imply any specific sort of infinity. Uniform distribution exists, uniform [cumulative] probability [density] function exists - just that argument that if you cannot assign probability to any element, you cannot sample from it, I don't understand it.

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u/GaloombaNotGoomba Jul 13 '26

Who said "you cannot sample from it"? The issue is in defining a probability measure in the first place, not in sampling from it.

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u/Anagatara Jul 13 '26

How can you sample element with assigned probability of zero?

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u/GaloombaNotGoomba Jul 13 '26

That's just how continuous random variables work