Two teams, Team A and Team B, will play in a playoff series against each other.
ID: 2928288 • Letter: T
Question
Two teams, Team A and Team B, will play in a playoff series against each other. The playoff series is composed of 7 separate games they play against each other. Team A has a 30% chance of winning each game. Let X be the number of games Team A wins.
(a) What is the distribution of X?
(b) In class we suggested a possible “representation” of X in terms of other random variables (call them Xi , say). What is this representation? Explain what the Xi are in terms of each of the 7 games.
(c) We know a formula for the mean of X. Use the representation to re-derive this formula.
(d) Let Y = n X. Show by the representation formula that Y Binomial(n, .7) and provide an interpretation for this result.
(e) A team wins the overall series by winning a majority (4) of the games. What is the probability that Team A wins the series? Does playing 7 games instead of just 1 game help or hurt Team A?
Explanation / Answer
(a) X is binomial distribution with parameter n=7 and p=0.3 , X~Binomial(n,0.3)
(b) P(X=x)=nCxpx(1-p)n-x
(c) mean(x)=np=7*0.3=2.1
(d) P(Y=n-X)=nCn-xp(n-x)(1-p)n-(n-x)=nCn-x(1-p)x)p(n-x) ,so Y=n-X is binomial distribution with parameter n=7 and 1-p=1-0.3=0.7 this imply Y~Binimal(n,0.7)
(e)P(winning a majority (4) of the games)=P(X>=4)=1-P(X<4)=1-P(X<=3)=1-0.874=0.126,
just 1 game help Team A,
as probability of winning a game =0.3 is more than probability of winning a series of 7 game=0.126.
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