(a) Define the conditional probability of the event given the event . Let be a partition of the sample space such that for all . Show that, if ,
(b) There are urns, the th of which contains red balls and blue balls. You pick an urn (uniformly) at random and remove two balls without replacement. Find the probability that the first ball is blue, and the conditional probability that the second ball is blue given that the first is blue. [You may assume that .]
(c) What is meant by saying that two events and are independent?
(d) Two fair dice are rolled. Let be the event that the sum of the numbers shown is , and let be the event that the first die shows . For what values of and are the two events independent?