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: 
The main issue with the question is that the density of primes decreases over increasing upper-limit interval subsets of $\mathbb N$. In fact, taking the density of the primes over the entire set of natural numbers evaluates to zero, therefore the probability would technically be zero. It does makes sense to say, for a given $n\in \mathbb N$, what is the probability of a number $q\in [1,n]$ being prime? And then the question has an exact, nonzero answer as the ratio of primes less than or equal to $n$ ($\pi(n)$) compared to $n$: $\pi(n)\over n$.
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).