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, <A HREF="https://books.google.nl/books/about/Prime_Numbers_and_Computer_Methods_for_F.html?id=5cIN7kemQgYC&redir_esc=y">Prime Numbers and Computer Methods for Factorization</A>, page 167 [first edition], page 159 [second edition].

[*] more precisely, this additive constant 1 should be replaced by $1.03465\ldots$ as calculated by <A HREF="http://dx.doi.org/10.1016/0304-3975(76)90050-5">Knuth and Trabb-Pardo</A> (appendix A); incidentally, if we don't count multiplicities the normal distribution has mean $0.26+\log(\log n)$ with the same standard deviation $[\log(\log n)]^{1/2}$, so the only difference is a slight displacement of the whole curve.