With two minutes left in a five-minute overtime, the score is 0-0 in a Rutgers soccer match versus Villanova. (That the overtime is NOT sudden-death.) In the next-to-last minute of the game, either (1) Rutgers scores a goal with probability p = 0.2, (2) Villanova scores with probability p = 0.2, or (3) neither team scores with probability 1 - 2p = 0.6. If neither team scores in the next-to-last minute, then in the final minute, either (1) Rutgers scores a goal with probability q = 0.3, (2) Villanova scores with probability q = 0.3, or (3) neither team scores with probability 1 - 2q = 0.4. However, if a team scores in the next-to-last minute, the trailing team goes for broke so that in the last minute, either (1) the leading team scores with probability 0.5, or (2) the trailing team scores with probability 0.5. For the final two minutes of overtime:

(a) Sketch a probability tree and construct a table for PR,V(r,v), the joint PMF of R, the number of Rutgers goals scored and V, the number of Villanova goals scored.

(b) What is the probability P[T] that the overtime ends in a tie?

(c) What is the PMF of R, the number of goals scored by Rutgers?

(d) What is the PMF of G, the total number of goals scored?