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.

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--- update 2020, in response to query:   

the "0.26" number is defined as
$$c_1= \gamma+\sum_{p\;\text{prime}}\biggl(\log(1-1/p)+\frac{1}{p}\biggr)= 0.261497212847643$$
while the "1.03" number is defined as
$$c_2=\gamma+\sum_{p\;\text{prime}}\biggl(\log(1-1/p)+\frac{1}{p-1}\biggr)= 1.034653881897438$$
The number $c_1$ is known as the <A HREF="https://en.wikipedia.org/wiki/Meissel–Mertens_constant">Meissel-Mertens constant.</A>
Both $c_1$ and $c_2$ are referred to as <A HREF="https://archive.lib.msu.edu/crcmath/math/math/h/h017.htm">Hadamard-de la Vallée-Poussin constants</A> (see also this <A HREF="http://mathworld.wolfram.com/MertensConstant.html">MathWorld entry).</A>