Seemingly you want to have two set [math] X_0, Y [/math] and know the size of the intersection [math] |X_0 \cap Y| [/math]

Now you want to assume that there is a random set [math] X [/math] which is a subset of a certain universe [math] S [/math] and conditional on [math] |X| = |X_0| [/math]

And now you want to find

[math] \Pr\{|X \cap Y| > |X_0 \cap Y|\} [/math]

If the above formulation is what you want, I think the question is quite advanced. You will need quite a bit of random set theory; it is because you need to define what do you mean by allowing a set distributed, let say "uniformly" inside a superset. After defining such probability measure we will have some knowledge to calculate the required probability. I can find a document like

http://www.cemmap.ac.uk/resources/molchanov_mc/lectures_sets.pdf
and I must admitted that I not touching these topics before. Maybe I am over-complicating the issues here.