It's been a while since I have studied statistics, but it seems I need it for my work.

I have found and area of intersection between two others on the earth. Now I would like to know if I took one of the areas and replaced it with an area of the same size, but in a totally random position, what is the probability of the new intersection being bigger than my old one. Or by this I mean the method I would use to do it.

It may be have written this badly, if I have just let me know and I'll rephrase the question.

Seemingly you want to have two set and know the size of the intersection

Now you want to assume that there is a random set which is a subset of a certain universe and conditional on

And now you want to find

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

I think what you're saying is right. I also know the size of |X| and |Y|. Also the set can be distributed uniformly. I have no experience of random set theory and assumed it maybe solvable using basic probability theory. But it appears that I am wrong. :/