Cramer's random model for the primes
Cramer's random model of the primes asserts, roughly speaking, that the primes behave as if every large integer n had an independent probability of 1 / logn of being prime (as predicted by the prime number theorem).
This model is not perfectly accurate, because it neglects some obvious structure in the primes, for instance the fact that they are mostly odd. However, these defects in the model can be repaired, for instance by resorting to the W-trick.
Cramer's random model predicts the Hardy-Littlewood prime tuples conjecture. Another prediction is that the number of primes in an interval [n,n + logn] for large generic n should asymptotically obey a Poisson distribution. This motivates Cramer's conjecture.