Let $X, Y$ be positive real numbers, and let $p_1, ... p_n$ be the primes less than $Y$. What is the probability that a product of distinct $p_i$'s is less than $X$, as a function of $X,Y$?

Clearly, if $X$ is sufficiently large relative to $Y$, then the probability is one. I am interested in the case when $Y$ is not too small compared to $X$, say $Y = X/C$ for some fixed positive constant $C$ larger than 1.