Problem 1:

(a) An urn contains four colors of balls, with three balls of each color. Three balls are removed randomly. What is the probability that only two different colors of balls are removed?

(b) Let X and Y be random variables where Var(X) =Var(Y) = 1. What are the minimum and maximum possible values of Var(X + Y)?

(c) Suppose that a medical test has a false-positive rate of 5%, meaning a healthy person will test positive with probability 5%, and the test has a false-negative rate of 3%, meaning that a sick person will test negative with probability 3%. Suppose a person being tested has a 2% chance of being sick. Given that the test result is positive, what is the probability that the person is sick?