## Probability unknown set definitions relate in function of intersection

Hello,

i would like to asses the probability of unknown set defenitions being related(or even identical) to each other, knowing only the elements in the sets and intersection. Does the probability the set defenitions are identical relate to the relationship between the elements in the intersection and the elements in the sets. Does the number of sets do anything on the probability the setdefenitions are identical?
What I try to do is evaluate the probability several syndroms(sets of symptoms) of unknown cause, but of known symptoms, having the same unknown cause. I would like to evaluate this probability starting from the relationship between the intersection and the different sets. Does a relatively big intersection rises the probability the syndromes share the same cause? (I ignore the symptoms might be a secod or third order process. For know I believe they result directly from the primary disease process).

Thank you for any insights.