Problem 2 [15 marks]

Consider the following situation:

There is an urn with 1 red ball and 1 blue ball.

Every time a ball is drawn (at random) from the urn, it is placed back in the urn along with 2 more balls of the same color, and 1 more ball of the other color.

Let Xi be the number of red balls after the i-th draw, for i = 1,2.... Note that Xi is the random variable representing the number of red balls in the urn including the three new balls added after the i-th draw.

(a) Find the probability mass function (pmf ) of X3. [5 marks]

(b) Compute E(X3) and V ar(X3). [5 marks]

(c) What is the probability that the first ball drawn was red, given that there are exactly 5 red balls after the third ball is drawn. [5 marks]

Problem 2 [15 marks] Consider the following situation: I There is an urn with 1 red ball and 1 blue ball. 0 Every time a ball is drawn (at random) from the urn, it is placed back in the urn along with 2 more balls of the same color, and 1 more ball of the other color. Let X; be the number of red balls after the i-th draw, for i = 1,2. . .. Note that Xi is the random variable representing the number of red balls in the urn including the three new balls added after the i-th draw. (a) Find the probability mass function (pmf) of X3. [5 marks] (b) Compute E(X3) and Var(X3). [5 marks] (0) What is the probability that the rst ball drawn was red, given that there are exactly 5 red balls after the third ball is drawn. [5 marks]