TITLE: Why does "the probability of a random natural number being prime" make no sense?
QUESTION [14 upvotes]: I read an essay about prime numbers. In it the author suggests that a natural first question to ask is, 

What is the probability of a random natural number being prime?

but proceeds to dismiss it as "not making sense". I wonder what's wrong with the question. Is its meaninglessness related to the fact that there is an infinite number of naturals?

REPLY [2 votes]: There is a sense in which the probability of a random natural number being prime approximates to $\frac{1}{log N}$, as Paul and Daniel each get to, and subject to Daniel's elegant exposition of the measure issue.
It's worth bearing in mind that for most of the twentieth century, a lot of mathematicians (and in particular, statisticians) burdened themselves with a very narrow definition of probability: that it represented the frequency of an occurrence in a long-enough series of independent identical trials.
Using that definition, it isn't really meaningful to ask whether a specific random natural number is prime or not: it either is (probability=1) or it is not (probability=0); and until you check, you don't know whether you are the former state or the latter. Every time you check that specific number, you will always get the same answer. So the estimate of $\frac{1}{log N}$ isn't meaningful (and nor is any other estimate, though if you choose 0, you'd be right more often than not), using that definition of probability.
There is a broader, more useful definition of probability: one that is a superset of the above definition. And that is, that it is the quantification of our reasoned belief that a specific event will occur. Under this definition, $\frac{1}{log N}$ is a meaningful estimate, and may be useful.