nt.number theory - Heuristic justification for Goldbach's conjecture

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.