Tennis Challenge In a recent U.S. Open tennis tournament, there were 879 challenges made by singles players, and 231 of them resulted in referee calls that were overturned. The accompanying table lists the results by gender.

	Challenge Upheld with Overturned Call	Challenge Rejected with No Change
Challenges by Men	152	412
Challenges by Women	79	236

a. If 1 of the 879 challenges is randomly selected, what is the probability that it resulted in an overturned call?

b. If one of the overturned calls is randomly selected, what is the probability that the challenge was made by a woman?

c. If two different challenges are randomly selected without replacement, find the probability that they both resulted in an overturned call.

d. If 1 of the 879 challenges is randomly selected, find the probability that it was made by a man or was upheld with an overturned call.

e. If one of the challenges is randomly selected, find the probability that it was made by a man, given that the challenge was upheld with an overturned call.

The number of calls that were overturned out of 879 challenges is listed. The values are categorizedaccording to the gender of the player.

a.

The total number of challenges is equal to 879.

The number of overturned calls is equal to\[152 + 79 = 231\].

The probability of selecting an overturned call is computed below:

\(\begin{array}{c}P\left( {{\rm{overturned}}\;{\rm{call}}} \right) = \frac{{{\rm{Number}}\;{\rm{of}}\;{\rm{overturned}}\;{\rm{calls}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{challenges}}}}\\ = \frac{{231}}{{879}}\\ = 0.263\end{array}\)

Thus, the probability of selecting an overturned call is equal to 0.263.

b.

The total number of overturned calls is equal to 231.

The number of calls made by women is equal to 79.

The probability of selecting an overturned call made by a woman is equal to:

\(\begin{array}{c}P\left( {{\rm{overturned}}\;{\rm{call}}\;{\rm{by}}\;{\rm{a}}\;{\rm{woman}}} \right) = \frac{{{\rm{Number}}\;{\rm{of}}\;{\rm{overturned}}\;{\rm{calls}}\;{\rm{made}}\;{\rm{by}}\;{\rm{women}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{overturned}}\;{\rm{calls}}}}\\ = \frac{{79}}{{231}}\\ = 0.342\end{array}\)

Thus, the probability of selecting an overturned call made by a woman is equal to 0.342.

c.

The probability of selecting two overturned calls without a replacement is computed below:

\[\begin{array}{c}P\left( {{\rm{both}}\;{\rm{overturned}}\;{\rm{calls}}} \right) = \frac{{231}}{{879}} \times \frac{{230}}{{878}}\\ = 0.0688\end{array}\]

Thus, the probability of selecting 2 calls without a replacement that were overturned is equal to 0.0688.

d.

The number of calls made by men is equal to:

\(152 + 412 = 564\)

The number of calls that were overturned is equal to 231.

The number of overturned calls made by a man is equal to 152.

Let A denote the event of selecting a call made by a man.

Let B denote the event of selecting an overturned call.

Thus, the probability of selecting a call that was made by a man or was overturned is computed below:

\(\begin{array}{c}P\left( {A\;or\;B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A\;and\;B} \right)\\ = \frac{{564}}{{879}} + \frac{{231}}{{879}} - \frac{{152}}{{879}}\\ = 0.732\end{array}\)

Thus, the probability of selecting a call that was made by a man or was overturned is equal to 0.732.

‘e.

Let A denote the event of selecting a call made by a man.

Let B denote the event of selecting an overturned call.

The probability of selecting a challenge that was made by a man given that it was upheld with an overturned call has the following expression:

\(P\left( {A|B} \right) = \frac{{P\left( {A\;and\;B} \right)}}{{P\left( B \right)}}\)

Now, the probability of selecting a call made by a man and was overturned is equal to:

\(\begin{array}{c}P\left( {A\;{\rm{and}}\;B} \right) = \frac{{{\rm{Number}}\;{\rm{of}}\;{\rm{overturned}}\;{\rm{calls}}\;{\rm{made}}\;{\rm{by}}\;{\rm{a}}\;{\rm{man}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{calls}}}}\\ = \frac{{152}}{{879}}\\ = 0.173\end{array}\)

The probability of selecting an overturned call is equal to 0.263 (refer to part a).

Now, the required probability is equal to:

\(\begin{array}{c}P\left( {A|B} \right) = \frac{{P\left( {A\;and\;B} \right)}}{{P\left( B \right)}}\\ = \frac{{0.173}}{{0.263}}\\ = 0.6577\\ \approx 0.658\end{array}\)

Therefore, the probability of selecting a challenge that was made by a man given that it was upheld with an overturned call is equal to 0.658.