First serve Tennis great Andy Murray made 60% of his first serves in a recent season. When Murray made his first serve, he won 76% of the points. When Murray missed his first serve and had to serve again, he won only 54% of the points. 21 Suppose you randomly choose a point on which Murray served. You get distracted before seeing his first serve but look up in time to see Murray win the point. What’s the probability that he missed his first serve?

Tennis Great first serve data In a recent season, Andy Murray said:

He made 60 percent of his first serves.

When he made first serve, he won 76 percent of the points.

When he missed his first serve and had to serve again, he won 54 percent of the points.

According to complement rule,

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

According to the general multiplication rule,

$P(\mathrm{A}\text{and}\mathrm{B})=P(\mathrm{A}\cap \mathrm{B})=P(\mathrm{A})\times P(\mathrm{B}\mid \mathrm{A})=P(\mathrm{B})\times P(\mathrm{A}\mid \mathrm{B})$

According to the addition rule for mutually exclusive events,

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

Definition for conditional probability:

$P(\mathrm{B}\mid \mathrm{A})=\frac{P(\mathrm{A}\cap \mathrm{B})}{P(\mathrm{A})}=\frac{P(\mathrm{A}\text{and}\mathrm{B})}{P(\mathrm{A})}$

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,

$P\left(F\right)=0.60$

Probability for made first serve wins point,

$P(\mathrm{W}\mid \mathrm{F})=0.76$

Probability for missed first serve wins point,

$P\left(\mathrm{W}\mid {\mathrm{F}}^{\mathrm{c}}\right)=0.54$

Now,

Apply the complement rule:

Probability for missed first serve,

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

Then

Apply general multiplication rule:

Probability for wins point and made first serve,

$P(\mathrm{W}\text{and}\mathrm{F})=P(\mathrm{F})\times P(\mathrm{W}\mid \mathrm{F})=0.60\times 0.76=0.456$

Probability for wins pointand missed first serve,

$P\left(\mathrm{W}\text{and}{\mathrm{F}}^{\mathrm{c}}\right)=P\left({\mathrm{F}}^{\mathrm{c}}\right)\times P\left(\mathrm{W}\mid {\mathrm{F}}^{\mathrm{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:

$P(\mathrm{W})=P(\mathrm{W}\text{and}\mathrm{F})+P\left(\mathrm{W}\text{and}{\mathrm{F}}^{\mathrm{c}}\right)$

$=0.456+0.216$

$=0.672$

Using conditional probability definition:

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

Thus,

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