Question

6. A and B are playing a short game of ping pong where A serves 3 times and B also serves 3 times. If after these six points one of them 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% of the 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 3 points when they serve (give the answers separately, so P(A wins 0 points when A serves)= , ...).

b) (4 pts) Find the probability that A scores a total of 4 or more 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 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) (4 pts) Find the probability that A wins if it goes to the second phase.

e) (2 pts) Find the probability that A wins (in either the first or second phase).

f) Extra credit (3 pts): find the expected number of points played.

Answer 1