I would try to be more clear in order that you can understand my calculations:

If an average user has 135 friends and the total facebook user is 500.000.000, and I want to make just an approximation without taking care about the dependency etc etc, I can state that the chance to be friend with a random user is 135/500.000.000.

Now to calculate with the same assumption what is the chance to have a mutual friend with a random user, I have to think that a mutual friend it's just a friend of a friend, so all the friends of my friends that are 135 X 135.

So if I want to approximate and I don't want take care about the dependency, I can state that the chance that a random guy is my mutual friend are (135^2)/500.000.000.

Starting from that, what I want to calculate now (with all the assumptions that you want) is: taking this random guy, and taking a set of n people, what is the chance that he is a mutual friend with at least 1 of them ?