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If I had a finite amount of red balls in a infinite sea of blue balls, what is the probability of grabbing a red ball? is it 0 because $\frac{n}{\infty} = 0$? But I can't accept that because if every one of those balls where grabbed by one of infinite people somebody would have needed to grab a red ball in order for all the balls to be grabbed

Dave
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    Probability zero does not always mean impossible. See for example https://math.stackexchange.com/questions/41107/zero-probability-and-impossibility – aschepler Feb 19 '24 at 01:08
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    Please repeat your question in your question body. Also in your body you probably mean $\frac{n}{\infty}$ and you should define what it means to pick a random ball out of infinite balls, since I'm not sure that is even well defined (cfr. Bertrand's paradox. – Vincent Batens Feb 19 '24 at 01:10
  • I did the first two things you suggested but I don't know exactly how to define grabbing a ball can you give a suggestion or two and I'll pick the best one? – Dave Feb 19 '24 at 01:18
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    As far as modern probability theory goes, there’s no way to define this probability, assuming the number of blue balls is countably infinite. There’s simply no probability measure on a countably infinite set that assigns the same probability to every element. – David Gao Feb 19 '24 at 02:46
  • What procedure did you have in mind for selecting a red ball from an infinite sea of blue balls at random? – Charles Hudgins Feb 19 '24 at 03:16
  • @CharlesHudgins I don't know maybe stick you hand in the sea without looking and see what comes up – Dave Feb 19 '24 at 04:40
  • @Dave Physically, it’s not possible to have an infinite number of blue balls. Even if it were possible, it’s impossible to be contained in a finite volume, so you can’t just “stick your hand in the sea” and pick at random, as your hand can only access a finite volume (and therefore only a finite number of balls) before grasping one of the balls. So really, the situation you describe is only an idealistic scenario. But as I mentioned, it is not a scenario describable in the language of modern probability theory (or any other mathematical theory, to my knowledge). – David Gao Feb 19 '24 at 05:06
  • The point of my comment is that you must accept that your physical intuition breaks down in nonphysical situations. You could have infinitely many people make infinitely many draws and still not exhaust the infinite sea of blue balls. Such is the nature of infinity. I can even explicitly construct a selection procedure where this is exactly what happens. – Charles Hudgins Feb 19 '24 at 13:37
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    To be concrete, enumerate the balls: $1, 2, 3, \ldots$. Suppose the first $N$ are red. Then in the first round person $1$ selects ball $p_1^{n_1}$ where $p_1$ is the first prime ($2$), and $n_1$ is the first positive power of $p_1$ greater than $N$. Person $k$ selects ball $p_1^{n_1 +k-1}$. In the $m$th round, person $k$ selects ball $p_m^{n_m + k-1}$. The infinity of primes and uniqueness of prime factorizations guarantees that we can perform infinitely many rounds of this selection procedure with infinitely many people, never selecting a red ball and never selecting a ball more than once. – Charles Hudgins Feb 19 '24 at 14:13

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Say there are r red balls and b blue balls. The probability that you grab a red ball is $\frac{r}{r+b}$. As $b$ tends to infinity, the probability tends to $0$.

Zachary J Kaplan
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