Question: Alice and Bob have just met, and wonder whether they have a mutual friend. Each has 50 friends, out of 1000 other people who live in their town. They think that it's unlikely that they have a friend in common, saying "each of us is only friends with 5% of the people here, so it would be very unlikely that our two 5%'s overlap." Assume that Alice's 50 friends are a random sample of the 1000 people (equally likely to be any 50 of the 1000), and similarly for Bob. Also assume that knowing who Alice's friends are gives no information about who Bob's friends are.
(a) Compute the expected number of mutual friends Alice and Bob have.
(b) Let X be the number of mutual friends they have. Find the PMF of X.
(c) Is the distribution of X one of the important distributions we have looked at? If so, which?