JJ and CC each have weighted coins: when flipped, these coins land on heads with probability 1/3 and tails with probability 2/3. Suppose JJ and CC each toss their coin twice. Let X be the number of heads JJ obtains and Y the number of heads CC obtains. In all of the problems below, show the calculation that leads to your answer.

a. We want to model this situation as a probability space. Thinking in terms of sets: what is our sample space S? What are the outcomes? What is the event X = 1?

b. What is the distribution of X? (In other words: for each value v in the range of X, what is P r[X = v]?) What is the distribution of Y ?

c. What is the distribution of X + Y ?

d. Given that X + Y = 2, find the distribution and expected value of X