Problem 3CRE
Tennis Challenge In the last U.S. Open tennis tournament, there were 611 challenges made by singles players, and 172 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 
121 
279 
Challenges by Women 
51 
160 
a. If one of the 611 challenges is randomly selected, what is the probability that it resulted in an overturned call?
b. If one of the challenges made by the men is randomly selected, what is the probability that it resulted in an overturned call?
c. If one of the challenges made by the women is randomly selected, what is the probability that it resulted in an overturned call?
d. If one of the overturned calls is randomly selected, what is the probability that the challenge was made by a woman?
e. If two different challenges are randomly selected with replacement, find the probability that they both resulted in an overturned call.
f. If one of the 611 challenges is randomly selected, find the probability that it was made by a man or was upheld with an overturned call.
g. If one of the challenged calls is randomly selected, find the probability that it was made by a man given that the call was upheld with an overturned call.
Answer :
Step 1 :
Given, In the last U.S. Open tennis tournament, there were 611 challenges made by singles players, and 172 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 
Total 
Challenges by Men 
121 
279 
400 
Challenges by Women 
51 
160 
211 
Total 
172 
439 
611 
 If one of the 611 challenges is randomly selected, we have to find the probability that it resulted in an overturned call.
There are 172 challenge upheld with overturned call
The total of challenges is 611.
Therefor,
the probability that it upheld with overturned call = 172/611 = 0.2815
b) If one of the challenges made by the men is randomly selected, we have to find the probability that it resulted in an overturned call.
Challenge Upheld with Overturned Call made by men = 121
The total of challenges is 611
Therefore,
the probability that it resulted in an overturned call = 121/611 = 0.198
c) If one of the challenges made by the women is randomly selected, we have to find the probability that it resulted in an overturned call.
Challenge Upheld with Overturned Call made by women = 51
The total of challenges is 611
Therefore,
the probability that it resulted in an overturned call = 51/611 = 0.0835
d) If one of the overturned calls is randomly selected, we have to find the probability that the challenge was made by a woman
There are 172 challenge upheld with overturned call
There are 51 challenges made by women that resulted in overturned.
Therefore,
the probability that the challenge was made by a woman = 51/172 = 0.2965
e) If two different challenges are randomly selected with replacement, we have to find the probability that they both resulted in an overturned call.
There are 172 challenge upheld with overturned call
The total of challenges is 611
Therefore,
the probability that they both resulted in an overturned call = = 0.0792
f) If one of the 611 challenges is randomly selected, we have to find the probability that it was made by a man or was upheld with an overturned call
The probability that it was made by man = = 0.655
the probability that it upheld with overturned call = 172/611 = 0.2815
The probability that challenge was made by a man and upheld by an overturned call is 121/611 = 0.198
the probability that it was made by a man or was upheld with an overturned call = 0.655+0.28150.198 = 0.7385
g) If one of the challenged calls is randomly selected, we have to find the probability that it was made by a man given that the call was upheld with an overturned call.
P(A/B) =
P(was made by a man / call was upheld with an overturned cal) =
= = = 0.7021