A firm has 30% of its service calls made by a contractor, and 15% of these calls result in customer complaints. The other 70% of the service calls are made by their own employees, and these calls have a 5% complaint rate. Use Bayes’ theorem to find the probability that a complaint was from a customer whose service was provided by their own employees.

E=calls by employees

C=complaints

P(E)=1-0.30=0.70
P(C|E')=0.15
P(C|E)=0.05
P(C)=0.7*0.05+0.3*0.15=0.08
P(E|C)=P(E∩C)/P(C)
=P(C∩E)/P(C)
=P(C|E)*P(E)/P(C) (Bayes' theorem)
=0.05*0.7/0.08
=35/80
=7/16
=

To solve this problem using Bayes' theorem, we need to calculate the probability that a complaint was from a customer whose service was provided by their own employees given the information provided.

Let's define the following events:
A: Complaint is from a customer whose service was provided by the firm's employees.
B: Complaint is made by a contractor.

We are given the following probabilities:
P(B) = 0.30 (30% of service calls are made by a contractor)
P(A|B) = 0.15 (15% of contractor service calls result in customer complaints)
P(A') = 0.70 (the complement of A, so 70% of service calls are made by the firm's employees)
P(A'|B) = 0.05 (5% of service calls made by a contractor result in customer complaints)

Now, we can use Bayes' theorem to find P(A|B'), which is the probability that a complaint was from a customer whose service was provided by the firm's employees given that the complaint was not made by a contractor.

Bayes' theorem states:
P(A|B') = (P(B'|A) * P(A)) / P(B')

We can calculate each term:
P(B') = 1 - P(B) = 1 - 0.30 = 0.70 (the probability that the complaint was not made by a contractor)
P(B'|A) = P(A'|B) * P(B) / P(A') = 0.05 * 0.30 / 0.70 = 0.02143 (the probability that the complaint was not made by a contractor given that the service was provided by the firm's employees)
P(A) = 1 - P(A') = 1 - 0.70 = 0.30 (the probability that the service was provided by the firm's employees)

Now, we can substitute the values into Bayes' theorem to find P(A|B'):
P(A|B') = (0.02143 * 0.30) / 0.70 = 0.0091714

Therefore, the probability that a complaint was from a customer whose service was provided by the firm's employees, given that the complaint was not made by a contractor, is approximately 0.0091714 or 0.92%.

To find the probability that a complaint was from a customer whose service was provided by their own employees, we can use Bayes' theorem. Bayes' theorem is a mathematical formula that relates conditional probabilities. It can be stated as:

P(A | B) = (P(B | A) * P(A)) / P(B)

Where:
P(A | B) is the probability of event A occurring given that event B has occurred.
P(B | A) is the probability of event B occurring given that event A has occurred.
P(A) is the probability of event A occurring.
P(B) is the probability of event B occurring.

Let's define the events:

A: A complaint was from a customer whose service was provided by their own employees.
B: A complaint was made.

We are given the following probabilities:
P(A) = 70% = 0.7 (probability of a service call made by their own employees)
P(B | A) = 5% = 0.05 (probability of a complaint given that the service call was made by their own employees)
P(B) = ? (probability of a complaint)

To find P(B), we need to consider the probabilities of complaints for each type of service call:

1. Contractor service calls:
P(complaint | contractor) = 15% = 0.15
P(contractor) = 30% = 0.3

2. Employee service calls:
P(complaint | employee) = 5% = 0.05
P(employee) = 70% = 0.7

The probability of a complaint can be calculated using the law of total probability:

P(B) = P(complaint | contractor) * P(contractor) + P(complaint | employee) * P(employee)
= 0.15 * 0.3 + 0.05 * 0.7
= 0.045 + 0.035
= 0.08

Now we can calculate the probability using Bayes' theorem:

P(A | B) = (P(B | A) * P(A)) / P(B)
= (0.05 * 0.7) / 0.08
= 0.035 / 0.08
= 0.4375

Therefore, the probability that a complaint was from a customer whose service was provided by their own employees is 0.4375 or 43.75%.