Chapter 7.3, Problem 55E

Tennis A single match at the men’s U.S. Open consists of a sequence of at most five sets that terminates when one person wins his third set. Suppose that the stronger person (the favorite) has probability p (where $p>\frac{1}{2}$ of winning any particular set. Then the probability of the weaker person (the underdog) winning any particular set is $1-p$.

(a) Explain why the probability that the underdog wins the match in five sets is $\left({}_2^4\right)p^2{\left(1-p\right)}^3$.

(b) Determine the probability that the underdog wins the match in three sets. In four sets.

(c) Show that, if $p=.7$, then the probability that the underdog wins the match is .16308.

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

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