54. Each member of a population is either type 1 with probability Image or type 2 with probabilityI mage. 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. LetI mage be the probability that a type i person will be friends with a randomly chosen other person.

(a) Find and Image.
LetI mage be the event that persons k and r are friends.

(b) Find Image.

(c) Show that Image.
Hint for (c): It might be useful to let X be such thatI mage,I mage.