Your roommate Dave eats one of two things for breakfast each day. Eighty percent of the time he has eggs for breakfast, and twenty percent of the time he has stale bread. After eating his breakfast he walks to class and for each person he sees, he randomly chooses to either wave hello nicely or scowl angrily. If he eats eggs then each time he sees a person he has a 90% probability of waving hello nicely, and each time he sees a person he has a 10% probability of scowling angrily. If he eats stale bread, then each time he sees a person he has a 60% probability of waving hello nicely, and each time he sees a person he has a 40% probability of scowling angrily. Dave's Greeting probability for each person by breakfast choice Breakfast choice Greeting Eggs Stale bread Wave nicely 90% for each person he sees 10% Scowl angrily 60% 40% (a) Dave eats breakfast and walks to class. On his way he sees n people. Derive an expression in terms of n and k for the probability that he will scowl angrily at k people and wave nicely to n k people. In other words find the following conditional probability: P(Scowls at k|Sees n people) (b) Suppose Dave sees n people on his way to class, waves nicely to n-k and scowls at k. Derive an expression for the probability he ate stale bread for breakfast. In other words, find the following conditional probability: P(Stale bread Saw n people, scowled at k)