Since you say that $X$ and $Y$ are events, let us rename them $A$ and $B$, to avoid a confusion with random variables.
Then, at least in the environmental, medical and life sciences literature, $P(A\cap B)/(P(A)P(B))$ is called the observed to expected ratio (abbreviation o/e). The idea is that the numerator is the actual probability of $A\cap B$ while the denominator is what it would be if $A$ and $B$ were independent.
Obviously the o/e ratio is $1$ if $A$ and $B$ are independent, it is more than $1$ if $A$ is favored by $B$, or, equivalently, if $B$ is favored by $A$, and it is less than $1$ if the opposite holds.
In the statistical analysis of genomic sequences, the CpGo/e ratio is especially important, which represents the frequency of the word CG divided by the product of the frequencies of the letters C (cytosine) and G (guanine), see here for an example. The rough idea is that in non functional portions of the genome, CpGo/e is much less than $1$ due to some well-known biological and chemical processes (a methylation-deamination of the guanine when it is right next to a cytosine, if you want to know). By contrast, in portions of the genome called CpG islands, CpGo/e is only slightly smaller than $1$ or even, greater than $1$, a fact which witnesses a repression of these processes and, as a consequence, may signal some functional regions.