I find that result fairly intuitive, when you understand how measure theory came up to be.

A much more surprising result is that most irrational are normal numbers, but we know almost no normal number (morally speaking, a normal number is an irrational number where each digit is equiprobable in any base).

Define "almost no". I mean, they have full measure and I can give countably many explicit examples, what more could you expect?

I think all mathematicians, given enough time, will eventually say countable to mean infinite.

This is virtually an axiom for continuous distributions.

One of the axioms of probability is that if you have an event (i.e. a set), then the probability of a countable union of disjoint sets is the sum of the probability of each set (event) occurring.

Assume a uniform distribution between 0 and 1. Now consider point sets of the rationals (i.e. the number 0.5 is represented by a set with just 0.5 in it). Since the distribution is uniform, each set has the same probability (i.e. the likelihood of picking a random rational).

Now consider this question: What is the probability of picking any rational between 0 and 1? Well, that's just the sum of the probabilities over all rationals (because it is a countable sum of disjoint sets). If the probability of picking any particular rational was non-zero, this sum would be infinite, which violates the laws of probability.

Thus, by convention, it's just simpler to define it to be 0.

There's no magic here. These properties were picked merely to make analysis with measure theory clean. Don't try to ascribe any real world meaning to picking a point.

I think this only sounds like a paradox if it is phrased poorly. The accurate way to state it is "The probability of randomly picking a specific number is 0" and that sounds reasonable. The probability of successfully picking any number is 1.

That's a different statement: OP is alluding to the fact the measure of Q is 0 when using the "standard" sigma algebra on the real line, while you are saying that the measure of a number of 0.

[edit] strictly speaking, you would restrict yourself to a bounded interval, e.g. if you pick a random number from a uniform distribution on [0, 1], the probability that this number is rational is 0.

oh, yeah, but that's because although Q is dense, it is not a dense subset of R and locally that's equivalent to saying a single point is not dense in R

no, it is not. The OP statement is about the probability of the event "{X in Q}" (equal to 0), with X a random variable uniformly distributed on a bounded interval. That even contains many points (infinitely many actually), but has a probability 0.

You are talking about the probability of a single point event, which is also always 0 on that same sigma algebra.

The OP point is not completely trivial because the event contains an infinite (but countably) number of elements. It is fairly easy to understand though since by its very definition, the P[{X in Q}] = sum P[{x}] taken over every rational number (since Q is countable), and each P[{x}] is 0.

A deeper statement is that there exists uncountable sets of probability 0.

The paradox is that, after picking a random number, you have just done a thing which has probability zero. Doing a thing that has zero probability shouldn't be possible. Ever.

You can't pick a random real number between 0 and 1. Heck, almost all reals between 0 and 1 can't ever be constructed let alone picked.

The here is the non-constructive nature of the real numbers. That is not to say the reals are useless, but they are not much more than a formalism. It's rather useful though because it's hard to get numbers like pi or e. Its really nice that any real interval is compact, but that too is hard to replicate.

You can certainly pick a random real number from the unit interval.

Really? Go on, then, pick one and tell us what it is (or at least tell us what your procedure was).

One example in a finite space and time setting would be selecting a random point on the ground. Say by dropping a ball there or something. The exact coordinates it lands is a random real number. But the probability that it landed on those exact coordinates is exactly 0, hence a paradox.

Non-countable sets defy intuition on several ways. The silver lining is that we don't have any evidence a non-countable thing exist on the real world.

I don't think anybody even has a procedure for gathering that kind of evidence.

No. Rational number are countable, but have the same property of zero probability for an item in a uniform distribution.

Why the downvotes? I think he is correct; and if someone think he is not, please elaborate

Yes, no reason for the downvotes.

Infinities are problematic too.

It is possible! Actually I remember​ pointing out a similar concern in my probability class back in the day. The teacher's answer: that is precisely the difference between probablity and possibility :)

I would argue that the distribution you used to pick a random number was not uniform. Not all the real number were equaliy likely to be picked by you. Hence the probability for some numbers was > 0.

Is it probability actually zero, or just infinitely close to zero?

If it's a random real then the probability is zero. If it's an arbitrarily close approximation to a random real then the probability is arbitrarily close to zero.

How can the sum of infinitely many zero probabilities be 1? I can understand how the sum of infinitely many infinitely close to zero values can be 1, but not infinitely many exactly zero values

There's no such thing as a "sum of infinitely many" anything. What we are talking about is the limit of an infinite series, which behaves nothing at all like a sum.

Those are the same thing. You're just calling them by different names.

Please describe how it is possible to pick such a number. For example, I can readily imagine how to pick a random 32b float, but that it is an entirely problem with a nonzero probability.

In probability theory, when dealing with continuous sample spaces / random variables, events with probability 0 still have a chance of occurring, and events with probability 1 stil l have a chance of NOT occurring, see:

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

This strange property comes from strange properties of the real numbers (and uncountably infinite sets) that give rise to things like:

https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

Measure theory deals with resolving this:

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

What's an anagram for Banach Tarski?

Banach Tarski Banach Tarski.

Seriously though, you can do math without invoking the axiom of choice. The formulation of probability doesn't strictly depend on it.

Usually in math we assume the axiom of choice :) https://en.wikipedia.org/wiki/Axiom_of_choice I'm assuming this could somehow lead to such a "random" pick in the technical sense.

In terms of implementation, I'm not aware of an algorithm that can randomly pick a real number on an actual computer. Perhaps a mathematician could show how to pick one on some abstract machine with infinite resources, and not constrained by finite bit representations of numbers.

> In terms of implementation, I'm not aware of an algorithm that can randomly pick a real number on an actual computer

An actual (finite in time and space) computer can't even represent arbitrary real numbers, much less randomly choose them.

A Turing machine can't pick random numbers of any kind.

Once you accept that you have an entropy source in the physical world, you can easily be injecting random real numbers (from some range) and in fact, usually are, which are then being binned into integers by ADCs.

What about PI? We can represent it in terms of "we know what we are talking about" and we can distinguish it from other numbers.

You can only have countable number of first or second order logic statements each defining a specific real number.

Those things sound equivalent, but I don't want to be the one to try to prove it.

Run a random number generator (0-9) for each decimal position after the dot in parallel. This should do a trick.

*entirely different

there are a number of similar paradoxes that arise when considering infinities of different sizes!

Infinity Paradoxes - Numberphile - https://www.youtube.com/watch?v=dDl7g_2x74Q

There is no uniform probability distribution on the reals.

Perhaps you meant the interval from 0 to 1?

There are many ways to pick randomly from the rationals with a non-uniform distribution.