Chapter 7.3, Problem 56E

Baseball The World Series consists of a sequence of at most seven games that terminates when one team wins its fourth game. Suppose that the stronger team (the favorite) the probability p (where $p>\frac{1}{2}$) of winning any particular game. Then the probability of the weaker team (the underdog) winning any particular game is $1-p$.

(a) Explain why the probability that the underdog wins the series in five games is $\left({}_3^4\right){\left(1-p\right)}^{4}p$.

(b) Determine the probability that the underdog wins the World Series in four games. Six games. Seven games.

(c) Show that, if $p=.6$, then the probability that the underdog wins the World Series is .289792.

(d) Explain why the probability in part (c) is the same as

$\left({}_0^7\right)p^0{\left(1-p\right)}^7+\left({}_1^7\right)p^1{\left(1-p\right)}^6+\left({}_2^7\right)p^2{\left(1-p\right)}^5+\left({}_3^7\right)p^3{\left(1-p\right)}^4$.