this variable $\Omega(n)$, the number of prime factors of $n$ counting multiplicity, has for large $n$ a normal distribution with mean [*] $1+\log(\log n)$ and standard deviation $[\log(\log n)]^{1/2}$; see, for example, Prime Numbers and Computer Methods for Factorization, page 167 [first edition], page 159 [second edition].
[*] more precisely, this additive constant 1 should be replaced by $1.03465\ldots$ as calculated by Knuth and Trabb-Pardo (appendix A); incidentally, if we don't count multiplicities the normal distribution has mean $0.26+\log(\log n)$ and the same standard deviation $[\log(\log n)]^{1/2}$