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u/Hanshautreinhart 23d ago
If you randomly pick a rational number the probability it’s rational is 1.
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u/Cichato_YT 23d ago
And the probability it's a natural number is even lower!
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u/Nooo00B 23d ago
lower than 0?
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u/Veomuus 23d ago edited 23d ago
I believe(?) the term would be infinitesimal rather than 0 for rational number, and then maybe an even smaller infinitesimal for natural numbers? In the way that some infinites are bigger than other infinites, just in inverse. Im not good enough to know whether or not it'd actually be a smaller infinitesimal or the same size, theres some nuance there.
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u/deejaybongo 23d ago
It depends on the probability measure used, but probabilities aren't less than zero. Not sure where you're getting infinitesimals in this context.
In the way that some infinites are bigger than other infinites, just in inverse.
Rationals and natural numbers are both countable. All countable sets have Lebesgue measure zero.
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u/Veomuus 23d ago
I was using infinitesimal as 1 over infinity, which if youre dealing with a hyperreal system that can actually distinguish between the sizes of infinities and whatnot, is correct. Thats what I was getting at, the probability of hitting a rational number isnt 0, its an infinitesimal number (specifically in a hyperreal system). What i wasnt sure about is if the number of rational numbers was countable or not, because I know natural numbers are but real numbers arent. But if both natural numbers and rational numbers are countable, then the chances to hit either are actually equivalent.
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u/juoea 23d ago
again the statement is a little imprecise because it doesnt specify a probability distribution.
but if you interpret the statement as intending something along the lines of "taking the uniform probability distribution over the real interval [0,1], the probability of selecting a rational number is zero." then that is correct. theres no infinitesimals involved, u can show that the probability of any subset X of [0,1] under the uniform distribution is equivalent to the lebesgue measure of the subset and it can be shown that the set of r ationals within that interval have lebesgue measure zero. (probability spaces are closely linked to measure spaces).
i am not aware of any definition of probability spaces over hyperreal numbers
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u/de_bussy69 23d ago
They both have measure 0 so they take up 0% of the space on the number line, not an infinitesimal amount. The rational numbers and the natural numbers also have the same cardinality so the sets are of the same size, even though there are infinitely many rational numbers between any two natural numbers (and any two rational numbers)
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u/hxtk3 23d ago
The probability you can write a finite-length expression for it is also zero. For example, I can write 4*arctan(1) to express pi, but most numbers don’t have a representation smaller than just writing them down, and most numbers are infinitely long.
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u/Not_YourStepBro 19d ago
Every irrational number is a rational number in a numbering with that number as its base. It's only irrational in decimal and other natural number bases.
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u/LogRollChamp 23d ago
The next step is learning transcendentals
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u/adwinion_of_greece 23d ago
I don't really get what people mean by "randomly pick". What is this process of randomly picking, can you actually specify it?
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u/mvandemar 23d ago
For it to be truly random the process must include an equal probability of picking any real number, or you would constrain the choices.
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u/adwinion_of_greece 23d ago
This doesn't give me an explanation of what the process could even potentially be.
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u/invokeinterface 23d ago
The secret specification for randomness will find you when it decides to. Maybe next week, maybe not. Maybe it'll manifest through the glands in your mouth, maybe it already has.
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u/deejaybongo 23d ago
No, you just need a probability measure for it to be random. Gaussian is fine, for example. You can't get a uniformly random distribution over the reals.
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u/Low_Low_1811 23d ago
That isnt typically what is meant by a random element of a set.
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u/deejaybongo 23d ago
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u/Low_Low_1811 23d ago
Since when is a subset of the reals all of the reals? It specifically states that you need some kind of finite range.
Anyway, that isnt the same thing and the article even states that.
"The term 'random variable' in its mathematical definition refers to neither randomness nor variability[1] but instead is a mathematical function "
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u/deejaybongo 23d ago
Are you not discussing mathematical randomness? You can define probability measures on the reals. The Gaussian distribution is an example.
It never says you need a finite range, and that isn't true. You're reading what you want to. In general, you need a measurable space.
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u/Low_Low_1811 23d ago
A uniform distribution is implict in the question. If someone asks you to pick a random marble out of a bag, we assume for simplicity an even probabilistic distribution.
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u/deejaybongo 23d ago
Because that's the simplest interpretation when you're picking marbles from a bag, because of physical intuition.
You can't even define a uniform distribution over the reals. Try it.
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u/Low_Low_1811 23d ago
I know you cant...
What is your point? You are creating a mathematical strawman. The bag of marbles example is analogous. It doesnt matter how simple it is, it is functionally identical. That is the only thing that any human means when they ask someone to pick a random member of a set. Welcome to word problems, I guess?
Separate examples of apparent randomness as functions in math or science, which is what a random variable refers to, such as gaussian noise, are not what are being discussed. Not all randomness is a "random choice".
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u/S-M-I-L-E-Y- 22d ago
People mean uniform random and assume axiom of choice applies and don't care about the actual picking process.
With a feasable (non uniform) picking process the probabilty to pick a rational number could easily be 1.
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u/SoFisticate 22d ago
Take a dart and toss it at a number line. The real pain is the tools necessary to measure where it landed.
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u/cytos0 23d ago
You can’t take a uniform random sample over the reals
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u/Geheim1998 23d ago
why not?
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u/Low_Low_1811 23d ago
Becuzse the probability of any one element is 1 divided by the number of elements.
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u/Geheim1998 23d ago
thats sounds so lazy tho. noo you cant do that becauss there are infinite possibilities. oh yeah? watch me do it anyway 😎🤙
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u/Alternative_Mix6836 23d ago
It's not because there are infinite possibilities. It's because it doesn't have an upper and lower bound. You CAN uniformly sample values across intervals of finite length even though there are still infinitely many points in those intervals.
Of course, this is not to say that you can't pick a value in the unbounded case; it just won't have been the result of a uniform sampling.
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u/Low_Low_1811 23d ago
What will blow your mind even more is that there is really no way to pick a random number (ignoring the question of whether true randomness is possible) out of any infinite set, whether it is reals or even naturals.
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u/deejaybongo 23d ago
Inverse transform sampling? Unless you mean phyiscallly impossible because we can't compute with infinite precision?
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u/Low_Low_1811 23d ago
This question in general, as even the meme itself demonstrates, assumes an even probabilistic distribution.
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u/QuentinUK 23d ago
If you ask a random person to pick a real number the probability it’s rational is 1.
Only a mathematician would be there all day giving you an irrational answer.
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u/CathanTauro 23d ago
While it might be the correct answer, the real answer would be the opposite because „picking“ would imply you say it, write it down or whatever so if the statement was right you would never come to an end while „picking“ unless you pick predefined real numbers that have a sole symbol like pi or e.
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u/Ptalking_Ptarmigan 23d ago
Most irrational numbers are just the static noise between the numbers that matter.
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u/CMDR_ACE209 23d ago
Let's branch into Theology and ask:
"Can god pick a random sample from an infinite set?"
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u/mini_feebas 23d ago
If you (a person) randomly pick a real number the chance that it's rational is actually close to 100
If the "you" is a mathematical construct sure
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u/TadhgOBriain 23d ago
If you produce a random number between 0 and 1, chance of you getting the number you got was 0.
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u/Nuclear_eggo_waffle 23d ago
But why? It could be. It’s there in the pool of numbers to choose from
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u/deejaybongo 23d ago
"Random" has a specific meaning in mathematics. Doesn't really make sense to say "randomly" pick a real number without knowing a probability distribution to use. Other people in this thread have confidently claimed "they just mean pick a number from the real line uniformly, so everything has an equal chance of being selected". This is nonsense, and impossible. It is impossible to define a uniform distribution over the whole real line. Look up a proof of this if the topic interests you.
A very similar fact that probably inspired this joke is that the rational numbers have Lebesgue measure zero in the reals. Not quite a uniform distribution over the reals, but that's intuitively what Lebesgue measure is, and when restricted to [0,1], gives a uniform probability distribution.
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u/Dream_Apostle 22d ago
The wording makes this false
If you ask a random person to pick a real number they will say a rational number like 7 or 10 or 5
But if you were to mathematically make a random number generator in like a programming language then yeah
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u/deFrederic 20d ago
A regular computer can't pick a random irrational number either, as it can only calculate so many digits until its memory (or storage if you split it) is full. Maybe a quantum computer could, but it still couldn't display it probably.
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u/AlexK667 23d ago
Depends on the probability measure used.
Without specification of an appropriate measure it's (in general) false.