Problem 37 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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?

Step-by-Step Solution

Answer

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

Step 1: 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

Step 2: 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

Step 3: 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.

Key Concepts

Bayes' theorem

Bayes' theorem is a powerful statistical tool used to update the probability of an event based on new evidence. It leverages prior knowledge or beliefs, in relation to the actual outcomes of an event. The theorem is instrumental in various fields such as medicine, finance, and engineering to make more informed decisions based on the likelihood of certain events.

For a more hands-on example, consider the Carter Temporary Help Agency case from the exercise. By using Bayes' theorem, we calculated the probability that an employee was placed by Darla given that their performance was unsatisfactory, which turned out to be around 64.71%. This probability helps the agency understand the performance of placements made by Darla and could impact future staffing decisions.

Bayes' theorem is mathematically expressed as:\[\begin{equation} P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}\end{equation}\]In plain terms, the theorem helps us determine the probability of event A occurring given that event B has occurred, by knowing the probabilities of A and B independently and the likelihood of B occurring if A happens.

Law of Total Probability

The Law of Total Probability is a fundamental rule relating to conditional probabilities that allows us to break down complex problems into simpler parts. The key benefit of this law is its ability to untangle the probability of a general event by considering all possible scenarios that could lead up to that event.

Applying this to our example with Carter Temporary Help Agency, we were concerned with the overall probability of an employee receiving an unsatisfactory rating, regardless of who placed them. Using the Law of Total Probability, we can calculate this overall likelihood by summing up the probabilities of an employee being unsatisfactory given they were placed by Nancy and by Darla, weighted by how often each of them places employees. It can be approached as a sum of individual scenarios that jointly cover all possibilities.

Mathematically, the Law of Total Probability for two events A and B is given by:\[\begin{equation}P(A)=P(A|B) \cdot P(B) + P(A|\overline{B}) \cdot P(\overline{B})\end{equation}\]Where \(P(\overline{B})\) is the probability of the complement of B occurring. In our problem, this provided a clearer picture of the overall job performance satisfaction levels across all employees.

Conditional Probability

Conditional Probability is a measure of the probability of an event occurring given the occurrence of another event. This concept is not just a theoretical tool; it's used to calculate probabilities in real-life situations when events are interlinked.

In the context of our Carter Temporary Help Agency example, conditional probabilities help us to understand the likelihood of employees being rated as unsatisfactory given they were placed by either Nancy or Darla. These probabilities reflect a deeper insight into the dynamics between employees' performances and the personnel responsible for their placement.

Conditional probability is expressed by the formula:\[\begin{equation}P(A|B) = \frac{P(A \cap B)}{P(B)}\end{equation}\]It essentially asks the question, 'If event B has happened, how likely is event A to happen?' This can guide better decision-making and predictive models when dealing with probabilistic events that are dependent on each other.