6. A and B are playing a short game of ping pong where A serves3 times and B also serves 3 times. If after these six points one ofthem is ahead the game ends, otherwise they go into a second phase.Suppose that A wins 70% of the points when they serve and 40% ofthe points when B serves.
Let’s look at the first phase.
a) (3 pts) Find the probability that A or B wins 0, 1, 2, or 3points when they serve (give the answers separately, so P(A wins 0points when A serves)= , ...).
b) (4 pts) Find the probability that A scores a total of 4 ormore points (so wins in the first phase).
c) (2 pts) Find the probability that A scores 3 points in total(so there is a tie in the first phase).
Now let’s look at cases where the game moves on to the secondphase. In this phase there are multiple rounds; in each round eachplayer serves once. They win if they win both points; otherwise itgoes to another round. Play continues until someone wins.
d) (4 pts) Find the probability that A wins if it goes to thesecond phase.
e) (2 pts) Find the probability that A wins (in either the firstor second phase).
f) Extra credit (3 pts): find the expected number of pointsplayed.
