Addressing the problem in its simplest form though, lets say that there is independence among the draws...that you don't have to worry about the correlation. Then this becomes the "same birthday" problem (look it up if you're interested). The probability that a randomly chosen person will have the same friend you do is just 1 minus the probability that the both of you share no friends.

This is very interesting, I didn't thought about it before: As the number of set of user increase the chance to find a mutual friend increase as well according to the birthday problem.

mmm how can I use the birthday paradox in my context ?

I want to formulate problem in another way to let you understand better:

I have a little network (say MyNet that is wrapped in Facebook) and I have an huge network ( Facebook), if there are only 2 user in MyNet (apparently with not relationship), what is the chance that exists a mutual friend in Facebook ?

When MyNet grows up say 1000 users, what is the chance that exist a mutual friend of any of 2 in facebook ?

How the chance increase according to the increasing of the MyNet's users ?

The birthday paradox seems to be a good point in this problem