  The goal of this paper is to describe an elementary combinatorial heuristic
that predicts Hardy and Littlewood's extended Goldbach's conjecture. We examine
common features of other heuristics in additive prime number theory, such as
Cram\'er's model and density-type arguments, both of which our heuristic draws
from. Apart from the prime number theorem, our argument is entirely elementary,
in the sense of not involving complex analysis. The idea is to model sums of
two primes by a hypergeometric probability distribution, and then draw
heuristic conclusions from its concentration behavior, which follows from
Hoeffding-type bounds.
