Chapter 5.3, Problem 83E

To determine

Conditional probability for Tennis great Andy Murray win point missed his first serve.

Answer to Problem 83E

Conditional probability that Tennis great Andy Murray win point missed his first serve is approx. 0.3214.

Explanation of Solution

Given information:

First serve data for Tennis Great Andy Murray in a recent season:

60% of his first serves he made.

76% of the points he won when made first serve.

54% of the points he won when missed first serve and had to serve again.

Calculations:

According to complement rule,

  $P\left({\text{A}}^{\text{c}}\right)=P\left(\text{not}\text{A}\right)=1-P\left(\text{A}\right)$

According to the general multiplication rule,

  $P\left(\text{A}\text{and}\text{B}\right)=P\left(\text{A}\cap \text{B}\right)=P\left(\text{A}\right)\times P\left(\text{B}|\text{A}\right)=P\left(\text{B}\right)\times P\left(\text{A}|\text{B}\right)$

According to the addition rule for mutually exclusive events,

  $P\left(\text{A}\cup \text{B}\right)=P\left(\text{A}\text{or}\text{}\text{B}\right)=P\left(\text{A}\right)+P\left(\text{B}\right)$

Definition for conditional probability:

  $P\left(\text{B}|\text{A}\right)=\frac{P\left(\text{A}\cap \text{B}\right)}{P\left(\text{A}\right)}=\frac{P\left(\text{A}\text{and}\text{B}\right)}{P\left(\text{A}\right)}$

Let

F: Made first serve

F^c: Missed first serve

W: Win point

W^c: Did not win point

Now,

The corresponding probabilities:

Probability for made first serve,

  $P\left(\text{F}\right)=0.60$

Probability for made first serve wins point,

  $P\left(\text{W}|\text{F}\right)=0.76$

Probability for missed first serve wins point,

  $P\left(\text{W}|{\text{F}}^{\text{c}}\right)=0.54$

Now,

Apply the complement rule:

Probability for missed first serve,

  $P\left({\text{F}}^{\text{c}}\right)=1-P\left(\text{F}\right)=1-0.60=0.40$

Then

Apply general multiplication rule:

Probability for wins point and made first serve,

  $P\left(\text{W}\text{and}\text{F}\right)=P\left(\text{F}\right)\times P\left(\text{W}|\text{F}\right)=0.60\times 0.76=0.456$

Probability for wins pointand missed first serve,

  $P\left(\text{W}\text{and}{\text{F}}^{\text{c}}\right)=P\left({\text{F}}^{\text{c}}\right)\times P\left(\text{W}|{\text{F}}^{\text{c}}\right)=0.40\times 0.54=0.216$

We know that

Andy Murray either missed his first serve or made his first serve.

Since both events are not possible at same time,

Apply addition rule for mutually exclusive events:

  $\begin{array}{l}P\left(\text{W}\right)=P\left(\text{W}\text{and}\text{F}\right)+P\left(\text{W}\text{and}{\text{F}}^{\text{c}}\right)\\ =0.456+0.216\\ =0.672\end{array}$

Using conditional probability definition:

  $P\left({\text{F}}^{\text{c}}|\text{W}\right)=\frac{P\left(\text{W}\text{and}{\text{F}}^{\text{c}}\right)}{P\left(\text{W}\right)}=\frac{0.216}{0.672}=\frac{9}{28}\approx 0.3214$

Thus,

Probability that the Tennis great Andy Murray win point missed his first serve is approx. 0.3214.