Chapter 4.2, Problem 48AYU

Challenge Problem Tennis Anyone? Assume that the probability of winning a point on serve or return is treated as constant throughout the match. Further suppose that $x$ is the probability that the better player in a match wins a set.

The probability $P_3$ that the better player wins a best-of-three match is $P_3=x^2\left[1+2\left(1-x\right)\right]$. Suppose the probability that the better player wins a set is $0.6$. What is the probability that this player wins a best-of-three match?

The probability Ps that the better player wins a best-of-five match is

$P_5=x^3\left[1+3\left(1-x\right)+6{\left(1-x\right)}^2\right]$

Suppose the probability that the better player wins a set is 0.6. What is the probability that this player wins a best-of-five match?

The difference between the probability of winning and losing is known as the win advantage. For a best-of- $n$ match, the win advantage is

$P_n-\left(1-P_n\right)=2P_n-1$

The edge, E, is defined as the difference: in win advantage between a best-of-five and best-of-three match. That is,

$E=\left(2P_5-1\right)-\left(2P_3-1\right)=2\left(P_5-P_3\right)$

Edge as function of win probability of set $x$ is

$E\left(x\right)=2x^2\left\{x\left[1+3\left(1-x\right)+6{\left(1-x\right)}^2\right]-\left(1+2\left(1-x\right)\right)\right\}$

Graph $E=E(x)$ for $0.5\le x\le 1$.

Find the probability of winning a set $x$ that maximizes the edge. What is the maximum edge?

Explain the meaning of $E(0.5)$?

Explain the meaning of $E(1)$?