Facebook friend probability

What is a probability to be friend with a random facebook user ?

Is it correct to state that, because the average friend of a Facebook user is 135 and the users are 500.000.000, the probability to be friend of a random user are 135/500.000.000, and is it still correct to say that the probability that I have a mutual friend with a random facebook users are (135*135)/500.000.000 ?

What happen if I want to calculate the probability to be mutual friend of at least one user taking n random users ?

Thank you to everyone can reply.


Can't make spagetti
this is a tad bit more complicated than you're appearing it to make. the probabilities of choosing a random user and it being a friend or not are not independent and, if i'm understanding you correctly, you're approaching them as such (although i have no clue as for where did you get the (135*135)/500.000.000, i'm pretty sure that's not right even if you assume that probabilities are independent). every time you sample from the population of users and see whether it is or it is not a friend your probabilities change (for instance, if you take the first user and discover that it is not a friend th next sample is going to be 499.999.999) although the hardest part here is that you're more likely to have a friend if you sample from a subset that's closer to you (same country, state/province, neighbourhood, etc...)
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 ?