The [heuristic justification](http://en.wikipedia.org/wiki/Goldbach%27s_conjecture#Heuristic_justification) section of the [Wikipedia article about Goldbach's conjecture](http://en.wikipedia.org/wiki/Goldbach%27s_conjecture) says that the argument that suggests that


*the number of twin primes below $x$ should be roughly $\dfrac{x}{\log^{2} x}$ up to a multiplicative constant*


is just heuristics, because of the lack of independence of the "random variables" considered therein. But what if we considered them as just **uncorrelated** instead of truly **independent**?


Could the argument be still used or not?




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Edit: several people have said that the notion of probability here is rather irrelevant, cause the primes form a deterministic system. My question now is thus: in some dynamical system framework, would the set of congruences that are verified in the case of primes force any subset of the positive integers with the same natural density as the one given by the PNT to be the one of a subset of the primes? The idea is to make a random process evolve under constraints until it becomes entirely determined. I apologize for the vagueness of the question, I just try to make my intuitions as 'rigorizable' as possible.
To make this more clear, let's consider the double sequence of random variables $X\_{i,t}$ with $1\leq i\leq n$ and $t\in\mathbb{N}\_{0}$ such that there exists $K\_{i,t}>0$ such that $$P(X\_{i,t}=1)=K\_{i,t}\dfrac{\pi(n)}{n}\left(\prod\_{k=1}^{t}(i \mod p\_{k})\right)^{1\_{t\gt 0}}$$
and $\forall t, \sum\_{i=1}^{n}P(X\_{i,t}=1)=1$. 



Primes as uncorrelated random variables
As I understand your question, we have random variables $X\_n$, $n\in\mathbb N$ taking values in $\{0,1\}$.


We intuitively think of $X\_n=1$ as ``$n$ is prime'', but other than that this has little to do with primes.


Letting $p\_n:=\Pr(X\_n=1)$, we have that heuristically $p\_n\rightarrow 0$ at a certain rate, and we want to conclude something about the number of $X\_k$'s that are $=1$ for $k\le n$.


Note that if $X\_i$ and $X\_j$ are uncorrelated, then they are (pairwise) independent. (This is true for any random variables that take values in $\{0,1\}$).


Moreover, the [Law of Large Nunbers only requires pairwise independence](https://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/).


[It may be that that fact breaks down for non-constant $\{p\_n\}\_{n\in\mathbb N}$ but I guess it should be okay.]


Thus, it seems the same informal argument **does** work with uncorrelatedness in place of independence.


