The MIT soccer team needs at least 8 players to avoid forfeiting a game. Assume

Question

The MIT soccer team needs at least 8 players to avoid forfeiting a game. Assume that each player has some chance of getting injured for the season, but her playing lifetime for a given season is exponentially distributed with parameter . For simplicity, assume that the coach insists on only playing 8 players at a time, and then replaces a player as soon as she gets hurt. Find:

(a) The expected time until the first substitution.

(b) The distribution of total time the team can play in a season, given that there are n women on the team.

Explanation / Answer

a) Lifetime of a woman for a given season =

P(a woman getting hurt or replaced is when collapses)=q= *e^- for x=1

X is the time until first substitution will follow geometric distribution q^xp where q is the prob of women getting hurt and replaced.

E(X)=q/p= *e^(- )/1- *(e^- )

b)Distribution of total time the team can play=(nx)p^x (1-p)^n-x where X is the total time team can play

p i=1-q where q has similar meaning as before.

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