Each member of a population is either type 1 with probability p1 or type 2 with probability p2 = 1 − p1. Independent of other pairs, two individuals of the same type will be friends with probability α, whereas two individuals of different types will be friends with probability β. Let Pi be the probability that a type i person will be friends with a randomly chosen other person.

(a) Find P1 and P2.

Let Fk,r be the event that persons k and r are friends.

(b) Find P(F1,2).

(c) Show that P(F1,2|F1,3) ≥ P(F1,2).

Hint for (c): It might be useful to let X be such that P(X = Pi ) = pi , i = 1, 2.