The probability that a randomly selected integer is squarefree

Question

What does

the probability that a randomly selected integer is squarefree

mean (in this context)?

Answer 1

It means this:

Let $n$ be a positive integer, and define $s_n$ to be the number of square-free integers between $1$ and $n$ inclusive. Then $$\lim_{n\to\infty}\frac{s_n}{n} = \frac{6}{\pi^2}$$

(Equivalently, you can allow negative integers, and define $t_n$ to be the number of square-free integers between $-n$ and $n$ inclusive. Then $\lim_{n\to\infty}\frac{t_n}{2n}= \frac{6}{\pi^2}$.)

Answer 2

Tony already explained the meaning... you take the "frequency" ("probability") of something happening on $[1,n]$, and then you take the limit as $n \to \infty$. This is not a probability in the sense of probability theory. There is no probability measure on a $\sigma$ - algebra on $\Bbb Z$ or $\Bbb N$ under which this number is the "probability" of an event. In fact no such probability space can exist. It is an unfortunate (confusing) abuse of terminology.

It is the same sense in which the "probability" of a randomly chosen integer being even (or odd) is 0.5.