TITLE: Heuristic justification for Goldbach's conjecture
QUESTION [9 upvotes]: On the Wikipedia page of Goldbach's conjecture, a heuristic justification is given, which did not completely satisfy me.  It roughly goes as follows:


randomly define a subset integers in accordance with the prime number
  theorem
Let $K_n$ be the random variable, counting the number of ways the
  natural number $2n$, can be written as
  a sum of two members of this set.

Then $E[K_n]\rightarrow \infty$ .

The problem is that, although the mean goes to infinity, it still might be true that the probability that $K_n>0$ for all $n$ is zero. 
So I thought of a different heuristic, and I am curious about whether anything is known about it:

Let $\mathcal P$ be the collection of
  all subsets of odd numbers whose
  density agrees with the prime number
  theorem, and let $\mathcal G$ be the
  collection of subsets for which
  Goldbach's property holds (i.e. every
  even number can be written in at least
  one way with two members of the set). 
  Let $\mu$ be the uniform product
  measure of the space $\{0,1\}^{\mathbb
> N}$.  Then the quantity  $$
> \frac{\mu(\mathcal P \cap \mathcal
> G)}{\mu(\mathcal P)} $$ 
   is (significantly) greater than zero. 
Edit: As pointed out in the comments,
  $\mu(\mathcal P) = 0$, so this
  quantity is  meaningless as it is, but
  I think it can be formalized in some
  way.

I do not know if this is easy or almost as difficult as the original problem.  But it would be a very convincing heuristic for me in that, it would tell me how much of Goldbach's conjecture is already explained by the prime number theorem.
I would appreciate answers, or references to any known results, or reasons if this kind of heuristic is not relevant, if that is the case.

REPLY [5 votes]: Let me add a historical remark. J.J. Sylvester was probably the first to come up with a heuristic argument.  See his (somewhat naive) approach in this short paper and this Nature note. 
P.S.  I found these two while working on this mildly related paper.