Question: I read an essay about prime numbers. In it the author suggests that a natural first question to ask is, 

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

Answer: 
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.
The probability that a random number n is prime is either 0 or 1; it will take us some time to find out which. In that sense probabilities make little sense.

But let’s take some large n, then count the primes p with n - n^1/2 < p < n + n^1/2, and calculate the probability that a random p in that range is a prime. The exact probability will depend on n, but it will be close to 1 / ln n, and the larger n gets the closer it gets to 1 / ln n.

So if I first pick a large n, then the probability that a random p in a large enough circle around n is prime is about 1 / ln n, depending on n and the size of the circle.

(This doesn’t work with any fixed size of the circle, we know there are intervals of size say 2,000,000 containing no primes at all and can even write them down. I haven’t proven that a size of n^1/2 is big enough but my guess is that it is).