We can use the Lindberg condition to show the distribution of number of prime divisors of an integer approaches Gaussian.

1. Is there a similar probabilistic formulation for square free numbers? That is, is it reasonable to say the probability of an uniformly random integer being square free is $\frac6{\pi^2}$ with suitable probabilistic interpretation?

2. Are there non-trivial and deceptive situations where such probabilistic interpretations break down?