# Basic Probability. [closed]

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 ?

• You can only speak of such probabilities if a probability measure has been given on $\mathbb N$. There is no uniform distribution on $\mathbb N$. Commented Dec 12, 2019 at 8:56
• This has been discussed recently in this answer Commented Dec 12, 2019 at 8:57
• @drhab. So no probability measure exists on the set of natural numbers. Is it true for all other infinite sets too ? Commented Dec 12, 2019 at 9:01
• 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. Commented Dec 12, 2019 at 9:04
• Makes sense now. Commented Dec 12, 2019 at 9:07

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.
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.