We have two urns, where urn-1 contains 3 blue balls and 1 red ball while urn-2 contains 2 blue balls and 2 red balls. Suppose we will randomly select one of the two urns, and then randomly pick a ball from the selected urn. a) What is the probability that we will pick a blue ball?
b) If we picked a blue ball, what is the probability that the selected urn was urn-1?
c) Suppose we picked a blue ball. If we randomly pick one additional ball from the same urn, what is the probability that we pick a red ball?

a) What is the probability that we will pick a blue ball?

there is a 50% chance that urn 1 will be selected and the possibility of a blue ball is 3/4 x 0.5 = 1.5/4

there is also a 50% chance that urn 2 will be selected and the possibility of a blue ball is 2/4 x 0.5 = 1/4

the probability of choosing a blue ball = 1.5/4 + 1/4 = 2.5/4 or 5/8

b) If we picked a blue ball, what is the probability that the selected urn was urn-1?

there are 5 blue balls in total, and 3 of them come from urn 1

c) Suppose we picked a blue ball. If we randomly pick one additional ball from the same urn, what is the probability that we pick a red ball?

there is a 50% chance that urn 1 will be selected, and the possibility of a red ball after a blue ball is 1/3 x 0.5 = 0.5/3

there is also a 50% chance that urn 2 will be selected and the possibility of a red ball after a blue ball is 2/3 x 0.5 = 1/3

the possibility of choosing a red ball after a blue ball = 0.5/3 + 1/3 = 1.5/3 = 1/2