Today, the professor brought an urn containing 3 balls to class. The professor tells you that there is a 50% chance that the urn is majority blue (2 blue balls, 1 red ball) and a 50% chance that the urn is majority red (1 blue ball, 2 red balls).

(a) Suppose that you draw from the urn once and get a blue ball. Given this information, what is the probability that the urn is majority blue?

(b) Suppose you place the ball you drew back in the urn and draw again. Now, you get another blue ball. Given that your two draws have both resulted in drawing a blue ball, what is the probability that the urn is majority blue?

(c) Now, suppose you again place the ball you drew back in the urn and draw one more time. This time you have drawn a red ball. What is the probability that the urn is majority blue, given the outcomes of your three draws?