A and B are playing a short game of ping pong where A serves 4 times and B serve

Question

A and B are playing a short game of ping pong where A serves 4 times and B serves twice. If after these six points one of them is ahead the game ends, otherwise they go into a second phase. Suppose that A wins 60% of the points when they serve and 30% of the points when B serves. Let's look at the first phase. a) Find the probability that A wins 0, 1, 2, 3, or 4 points when they serve and 0, 1, or 2 points if B serves (give the answers separately, so P(A wins 0 points when A serves) =, ..). b) Find the probability that A scores a total of 4 or more points (so wins in the first phase). c) 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) 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 points played.

Explanation / Answer

Probability that A wins when he serves, P(A) = 0.6
Probability that B wins when he serves, P(B) = 0.3

(A)
P( A wins 0 points when A serves) = (1 - 0.6)^4 = 0.4^4 = 0.0256

P(A wins 1 point when A serves) = 4C1*0.6^1*0.4^3 = 0.1536

P(A wins 2 point when A serves) = 4C2*0.6^2*0.4^2 = 0.2304

P(A wins 3 point when A serves) = 4C3*0.6^3*0.4^1 = 0.3456

P(A wins 4 point when A serves) = 4C4*0.6^4*0.4^0 = 0.1296

P(B wins 0 point when B serves) = 2C0*0.3^0*0.7^2 = 0.49

P(B wins 1 point when B serves) = 2C1*0.3^1*0.7^1 = 0.42

P(B wins 2 point when B serves) = 2C2*0.3^2*0.7^0 = 0.09

(B)

P(A wins) = P( A wins 4 points when A serves)*(P(B wins 2 point when B serves) + P(B wins 1 point when B serves) + P(B wins 0 point when B serves)) + P(A wins 3 points when A serves)*(P(B wins 1 points when B serves) + P(B wins 0 points when B serves)) + P(A wins 2 point when A serves)*(P(B wins 0 points when B serves)) = 0.1296*(0.09 + 0.42 + 0.49) + 0.3456*(0.42 + 0.49) + 0.1536*0.49 = 0.12467

(C)
P(A scores 3 points) = P(A wins 1 point when A serves)*P(B wins 0 point when B serves) + P(A wins 2 point when A serves)*P(B wins 1 point when B serves) + P(A wins 3 point when A serves)*P(B wins 2 point when B serves) = 0.1536*0.49 + 0.2304*0.42 + 0.3456*0.09 = 0.203136

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