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What is the probability that l get a certain whole number, say 2 out of the set of the natural numbers. Is it zero ?

Further more, what is the probability of obtaining an even number, Is it 1/2 ?

Is it meaningful to define probability over an infinite set or not ?

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    $\begingroup$ You can only speak of such probabilities if a probability measure has been given on $\mathbb N$. There is no uniform distribution on $\mathbb N$. $\endgroup$
    – drhab
    Commented Dec 12, 2019 at 8:56
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    $\begingroup$ This has been discussed recently in this answer $\endgroup$
    – Arnaud Mortier
    Commented Dec 12, 2019 at 8:57
  • $\begingroup$ @drhab. So no probability measure exists on the set of natural numbers. Is it true for all other infinite sets too ? $\endgroup$
    – Aristotle Stagiritis
    Commented Dec 12, 2019 at 9:01
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    $\begingroup$ I am not saying that no probability measure exists on $\mathbb N$ but that no uniform probability measure (i.e. one such that all numbers have equal probability to be chosen) exists. On every non-empty set (also infinite) we can define probability measures. The non-existence of a uniform probability measure implicitly says that we cannot use terminology like "a randomly drawn number from $\mathbb N$". This terminology is senseless for infinite sets. $\endgroup$
    – drhab
    Commented Dec 12, 2019 at 9:04
  • $\begingroup$ Makes sense now. $\endgroup$
    – Aristotle Stagiritis
    Commented Dec 12, 2019 at 9:07

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Defining probability over an infinite set is tricky. You cannot do it so that all singletons have equal probability, else it would be zero.

Thus you need to define a density of probability in order to state which is the probability of having a certain result. In $\mathbb N$ there is no probability measure that is uniform.

But you could give meaning to choosing at random by fixing a probability of choosing each number.

E.g.: $P(n)=\frac{1}{2^{n+1}}$ is a possibility: you have one possibility in two of choosing $0$, one in four of choosing $1$ and so on. For this choice, $P(even)=2 P(odd)$ hence $P(even)=\frac{2}{3}$.

Obviously you can give a different distribution of probability and get a different result. The point is: since $\mathbb N$ is infinite, it will not be uniform.

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  • $\begingroup$ Interesting, it seems to be quite flexible and allows different measures. $\endgroup$
    – Aristotle Stagiritis
    Commented Dec 12, 2019 at 9:16
  • $\begingroup$ Indeed it is. Actually on any space you can fix a probability measure such that some basic rules apply (see en.wikipedia.org/wiki/Probability_measure) and then various theorems of probability apply. The choice of probability measure is a pre-mathematic problem, i.e. you have to chose what it does mean randomly choose in your situation. Uniform probability for a finite set of singletons is just a simple case of the theory. $\endgroup$
    – Alberto Saracco
    Commented Dec 12, 2019 at 14:02

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