The probability that an integer $p$ divides an integer $x$ is $\dfrac{1}{p}$.
From this article on almost prime numbers, the number $\pi_k(n)$ of positive integers less than or equal to $n$ with at most $k$ divisors (not necessarily distinct) is asymptotic to:
$$\pi_k(n) \sim \left(\frac{n}{\log n}\right)\frac{(\log \log n)^{k-1}}{(k-1)!}$$
Is there a way to estimate the number of positive integers less than or equal to $n$ that have a given prime $p$ as the least prime factor.