Suppose there are three urns. Urn 1 contains two red balls and two black balls. Urn 2 contains one red ball and three black balls. Urn 3 contains two red balls and one black ball. A ball is drawn at random from Urn 3. If it is red, a second ball is drawn at random from Urn 2, but if it is black, the second ball is drawn at random from Urn 1.

a) What is the probability of the second ball being drawn from Urn 2?

b) What is the probability that the second ball is black, given that it is drawn from Urn 2?

c) What is the probability that the second ball is black?

d) What is the probability that the second ball was drawn from Urn 1, given that it is black?

e) What is the probability that the first ball drawn was red, given that the second ball drawn is black?

f) What is the probability that both balls drawn are red?