A large firm has 85% of its service calls made by a contractor, and 10% of these calls
result in customer complaints. The other 15% of the service calls are made by their
own employees, and these calls have a 5% complaint rate. Find the
(a) probability of receiving a complaint.
(b) probability that the complaint was from a customer serviced by the
contractor.
A) To find the probability of receiving a complaint, we need to calculate the weighted average of the complaint rates for calls made by the contractor and the employees.
Let's denote:
C1 = the probability of a complaint for calls made by the contractor = 10%
C2 = the probability of a complaint for calls made by the employees = 5%
P1 = the probability of a call being made by the contractor = 85%
P2 = the probability of a call being made by the employees = 15%
The probability of receiving a complaint is given by:
P(complaint) = P1 * C1 + P2 * C2
= 0.85 * 0.10 + 0.15 * 0.05
= 0.085 + 0.0075
= 0.0925
≈ 9.25%
So, the probability of receiving a complaint is approximately 9.25%.
B) To find the probability that the complaint was from a customer serviced by the contractor, we need to calculate the conditional probability of the complaint given that the call was made by the contractor.
Let's denote:
P(complaint | contractor) = the conditional probability of a complaint given a call made by the contractor
P(complaint) = the probability of a complaint (calculated in part A)
The probability that the complaint was from a customer serviced by the contractor is given by:
P(contractor | complaint) = P(complaint | contractor) * P(contractor) / P(complaint)
= (0.10 * 0.85) / 0.0925
= 0.085 / 0.0925
≈ 0.9189
≈ 91.89%
So, the probability that the complaint was from a customer serviced by the contractor is approximately 91.89%.
To find the probability of receiving a complaint, we need to calculate the weighted average probability of receiving a complaint for both the contractor and employees.
(a) Probability of receiving a complaint:
Let's denote C as the event of receiving a complaint.
P(C) = P(C|Contractor) * P(Contractor) + P(C|Employee) * P(Employee)
P(C) = (0.10 * 0.85) + (0.05 * 0.15)
P(C) = 0.085 + 0.0075
P(C) = 0.0925
Therefore, the probability of receiving a complaint is 0.0925 or 9.25%.
(b) Probability that the complaint was from a customer serviced by the contractor:
Let's denote D as the event of the complaint being from a customer serviced by the contractor.
P(D|C) = P(C|Contractor) * P(Contractor) / P(C)
P(D|C) = (0.10 * 0.85) / 0.0925
P(D|C) = 0.085 / 0.0925
P(D|C) ≈ 0.9189
Therefore, the probability that the complaint was from a customer serviced by the contractor is approximately 0.9189 or 91.89%.
To find the probability of receiving a complaint, we need to consider two scenarios: a complaint from a customer serviced by the contractor and a complaint from a customer serviced by the firm's own employees.
(a) Probability of receiving a complaint:
Let's denote the event of a complaint as C and the event of a service call as S.
P(C) = P(C from contractor) + P(C from firm's own employees)
P(C from contractor) = probability of a complaint given that the call was made by the contractor = 10% = 0.10
P(C from firm's own employees) = probability of a complaint given that the call was made by the firm's own employees = 5% = 0.05
Now, we need to consider the probability of a service call made by the contractor and the firm's own employees.
Let's denote the event of a service call made by the contractor as Cc and the event of a service call made by the firm's own employees as Ce.
P(Cc) = probability of a service call made by the contractor = 85% = 0.85
P(Ce) = probability of a service call made by the firm's own employees = 15% = 0.15
Using these probabilities, the probability of receiving a complaint can be calculated as:
P(C) = P(C from contractor) * P(Cc) + P(C from firm's own employees) * P(Ce)
P(C) = 0.10 * 0.85 + 0.05 * 0.15
Simplifying this equation, we get:
P(C) = 0.085 + 0.0075
P(C) = 0.0925
Therefore, the probability of receiving a complaint is 0.0925 or 9.25%.
(b) Probability that the complaint was from a customer serviced by the contractor:
To find this probability, we need to consider the probability of a complaint given that the call was made by the contractor and divide it by the probability of receiving a complaint calculated in part (a).
Let's denote the event of a complaint from a customer serviced by the contractor as Cc.
P(Cc) = probability of a complaint given that the call was made by the contractor = 10% = 0.10
To find the probability that the complaint was from a customer serviced by the contractor, we can use the conditional probability formula:
P(Cc | C) = P(Cc ∩ C) / P(C)
P(Cc | C) = P(Cc) * P(Cc) / P(C)
P(Cc | C) = 0.10 * 0.85 / 0.0925
Simplifying this equation, we get:
P(Cc | C) ≈ 0.9189
Therefore, the probability that the complaint was from a customer serviced by the contractor is approximately 0.9189 or 91.89%.
Lets say C stands for done by contractor, and F stands for done by the firm itself. N stands for negative (complained calls), and P stands for positive (uncomplained calls). Thus, NsubC is complained contractor calls, NsubF is complained firm calls, PsubC is uncomplained contractor calls, and PsubF is uncomplained firm calls.
Total contractor calls = 85% = PsubC + NsubC = (1)*85% = (90% + 10%)*85%
NsubC = 10%*85% = (1/10)*85% = 8.50%
Total firm calls = 15% = PsubF + NsubF = (1)*15% = (95% + 5%)*15%
NsubF = 5%*15% = (1/20)*15% = 0.75%
N = NsubC + NsubF = 8.50% + 0.75% = 9.25%
∴
a.) N = 9.25%
b.) NsubC = 8.50%