An urn contains 5 blue and 7 gray balls. Let us say that 2 are chosen at random, one after the other, without replacement.
a. Find the following probabilities and illustrate them with a tree diagram: the probability that both balls are blue, the probability that the first ball is blue and the second is not blue, the probability that the first ball is not blue and the second ball is blue, and the probability that neither ball is blue.
b. What is the probability that the second ball is blue?
c. What is the probability that at least one of the balls is blue?
d. If the experiment of choosing two balls from the urn were repeated many times over, what would be the expected value of the number of blue balls?