The probability that a random number N has at least M factors That is, how to calculate it given the size of N(that is, logN) and assuming that logN is much greater than M. Its an approximation. There is no exact formula. 
I do know that according to the prime number theorem the probability for N being prime is 1/ln(N) for large positive integers. 
Thanks.
 A: The count of numbers up to $x$ having at most $k$ prime divisors (counted with mutiplicity) is 
asymtotically equal to (see for example Almost prime on Wikipedia)
$$ \frac{x}{\log x} \frac{(\log \log x)^{k-1}}{(k-1)!} .$$
Thus, since the number of divisors is definitely greater than the number of prime divisors, the probabilty (in the informal sense used in the question) that some $N$ has at least $M\ge 3$ divisors is (asymtotically) at least 
$$ 1 - \frac{1}{\log N} \frac{(\log \log N)^{M-3}}{(M-3)!} .$$
It might well be possible to improve this as there are 'few' numbers for which the number of primes divisors (with mutiplicty) is 'close to' the number of divisors. 
A quick guess, possibly wrong, would be that one might be able to improve this towards something like
$$ 1 - c \frac{(\log \log N)^{\log M}}{\log N}, $$ by looking at distinct prime divisors.
And this would be about optimal, which can be seen via noting that if the number of prime divisors (with multiplicty) is $k$ this bounds the number of divisors by $2^k$, and using the above argument in the other direction. 
A: The Erdős-Kac theorem gives that, for a fixed function $M(N)$ with $\limsup M(N)/\log \log N < 1$, 
$\frac{1}{N}  |\{n\in[N,2N] : \omega(n) > M(N)\}|\rightarrow 1$.
Likewise if $\liminf M(N)/\log \log N > 1$, 
$\frac{1}{N}  |\{n\in[N,2N] : \omega(n) > M(N)\}| = o(1)$,
where here $\omega(n)$ counts the number of prime divisors of $n$ without multiplicity. (So $\omega(4) = \omega(2) = 1$, while $\omega(6) = 2$.) The same results will be true if prime factors are counted with multiplicity however. (i.e. we consider $\Omega(n)$, where $\Omega(4) = 2$ for instance.)
More precise asymptotics can be obtained, especially easily for $M(N) = o(\log \log N)$, by using formula of Sathe and Selberg, and its extensions. These are uniform versions of the theorem of Landau which has been mentioned by quid. Where $M(N)$ grows like $\log \log N$ or faster, I'm afraid these formula become somewhat complicated, and I wouldn't expect a nice asymptotic expression (but I could be wrong). A reference is "On the number of prime factors of an integer" by Hildebrand and Tenenbaum, the easiest offshoot of which (due to Sathe) is that Landau's formula holds uniformly for $k = o(\log \log x)$. Formula (1.7) of Pomerance will give you (with a little patience) nice upper bounds.
The book of Tenenbaum already mentioned is also nice reference for some of these questions, as is chapter 7 of Montgomery and Vaughan's "Multiplicative Number Theory I". Kac's book is great for anyone to read, interested in these questions or not.
A: For all such questions ( including the meta question of what the question really is) check out Gerald Tenenbaum's analytic and probabilistic number theory book. The book certainly talks at length about the statistics of the divisor function.
