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Chapter 7: Problem 37

Applicants for temporary office work at Carter Temporary Help Agency who have successfully completed a typing test are then placed in suitable positions by Nancy Dwyer and Darla Newberg. Employers who hire temporary help through the agency return a card indicating satisfaction or dissatisfaction with the work performance of those hired. From past experience it is known that \(80 \%\) of the employees placed by Nancy are rated as satisfactory, and \(70 \%\) of those placed by Darla are rated as satisfactory. Darla places \(55 \%\) of the temporary office help at the agency and Nancy the remaining \(45 \%\). If a Carter office worker is rated unsatisfactory, what is the probability that he or she was placed by Darla?

Short Answer

Expert verified
If a Carter office worker is rated unsatisfactory, there is approximately a \(64.71\%\) chance that he or she was placed by Darla.

Step by step solution

01

Identify the probabilities

We are given the following probabilities: 1. Probability of an employee placed by Darla: P(D)=0.55 2. Probability of an employee placed by Nancy: P(N)=0.45 3. Probability of an employee being satisfactory if placed by Darla: P(S|D)=0.7 4. Probability of an employee being satisfactory if placed by Nancy: P(S|N)=0.8 Now, we can calculate the probability of being unsatisfactory given they were placed by Darla or by Nancy: 1. P(U|D)=1-P(S|D)=1-0.7=0.3 2. P(U|N)=1-P(S|N)=1-0.8=0.2
02

Calculate the probability of unsatisfactory ratings

Now we want to find the probability of an employee being rated unsatisfactory in general, which we can call P(U). We can use the Law of total probability to find this value. P(U)=P(U|N)*P(N)+P(U|D)*P(D) P(U)= 0.2*0.45 + 0.3*0.55 P(U)= 0.09 + 0.165 P(U)= 0.255
03

Use Bayes' theorem

Now we can apply Bayes' theorem to find the probability that an employee rated unsatisfactory was placed by Darla, which we can write as P(D|U). P(D|U)= P(U|D)*P(D) / P(U) P(D|U)= (0.3*0.55) / 0.255 P(D|U)= 0.165 / 0.255 P(D|U)= 0.6471 (approximately) So, if a Carter office worker is rated unsatisfactory, there is approximately a 64.71% chance that he or she was placed by Darla.

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