Question

Problem 4. Two hockey teams, A and B play a series of games, until one of the teams wins 4 games. Suppose team A has probability p of winning each game, and games are independent. Let X be the total number of games that are played (a) Find the probability mass function of X (b) What is the probability that team A wins the series conditioned on X-47 (c) What is the probability that team A wins the series conditioned on X 7? (Simplify your expressions as much as possible.)

Answer 1

a)pmf of X is as follows:

P(X=4)=P(A wins first 4 games)+P(B wins 4 first games)=p⁴+(1-p)⁴

P(X=5)=P(A wins 3 of first 4 and wins 5th game)+P(B wins 3 of first 4 and wins 5th game)

=⁴C₃p⁴(1-p)¹+⁴C₃p¹(1-p)⁴ =4*p*(1-p)*(p³+(1-p)³)

P(X=6)=⁵C₃p⁴(1-p)²+⁵C₃p²(1-p)⁴ =10*p²(1-p)²*(p²+(1-p)²)

P(X=7)=⁶C₃p⁴(1-p)³+⁶C₃p³(1-p)⁴ =20*p³(1-p)³*(p+(1-p))=20*p³(1-p)³

b)

P(Awins |X=4)=P(A wins first 4 games)/P(X=4)=p⁴/(p⁴+(1-p)⁴)

c)

P(Awins |X=7)=⁶C₃p⁴(1-p)³/(20*p³(1-p)³) =p