A and B are playing a short game of ping pong where they each serve twice. If after these four points one of them is ahead by at least two points the game ends, otherwise they go into a second phase. Suppose that A wins 70% of the time when they serve and 40% of the time when B serves. Let?s look at the first phase.
a) Find the probability that A scores 0, 1, or 2 points when they serve and when B serves.
b) Find the probability that A scores a total of 3 or 4 points (so wins in the first phase).
c) Find the probability that A scores 2 points in total (so there is a tie in the first phase).
Now lets look at cases where the game moves on to the second phase. In this phase there are multiple rounds: in each round each player serves once. They win if they win both points; otherwise it goes to another round. Play continues until someone wins.
d) Find the probability that A wins if it goes to the second phase.
e) Find the probability that A wins (in either the first or second phase).
f) Find the expected number of point s played.