their forms. rest. 11) An insurance company subcontracts two data processing firms to handle Firm #1 handles 80% of the forms and Firm #2 handles the not perfect: it is known that 2% of all forms processed by Firm #1 Unfortunately, these firms are run by people and are therefore will be processed incorrectly and that 5% of those from Firm #2 will also be processed incorrectly. Suppose a random form is chosen from those that have been processed. What is the probability that this form was processed by Firm #2, given that it was processed incorrectly? (Hint: a probability tree or Bayes' Theorem may be
their forms. rest. 11) An insurance company subcontracts two data processing firms to handle Firm #1 handles 80% of the forms and Firm #2 handles the not perfect: it is known that 2% of all forms processed by Firm #1 Unfortunately, these firms are run by people and are therefore will be processed incorrectly and that 5% of those from Firm #2 will also be processed incorrectly. Suppose a random form is chosen from those that have been processed. What is the probability that this form was processed by Firm #2, given that it was processed incorrectly? (Hint: a probability tree or Bayes' Theorem may be
Chapter6: Systems Of Equations And Inequalities
Section: Chapter Questions
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![### Probability and Data Processing: A Case Study of Subcontracted Work
**Problem Statement:**
An insurance company subcontracts two data processing firms to handle their forms. Firm #1 handles 80% of the forms, while Firm #2 handles the rest. Unfortunately, these firms are run by people and are therefore not perfect. Here are some key details:
- It is known that 2% of all forms processed by Firm #1 will be processed incorrectly.
- 5% of the forms from Firm #2 will also be processed incorrectly.
**Question:**
Suppose a random form is chosen from those that have been processed. What is the probability that this form was processed by Firm #2, given that it was processed incorrectly? (Hint: use a probability tree or Bayes' Theorem for the solution.)
**Detailed Explanation:**
When solving this problem, it's helpful to use Bayes' Theorem, which helps us revise existing predictions based on new evidence. Also, we can use a probability tree to visualize the problem.
### Information Recap
- Probability a form is handled by Firm #1: P(F1) = 0.80
- Probability a form is handled by Firm #2: P(F2) = 0.20
- Probability of incorrect processing by Firm #1: P(I|F1) = 0.02
- Probability of incorrect processing by Firm #2: P(I|F2) = 0.05
Our task is to find the probability that a form was handled by Firm #2 given that it was processed incorrectly, i.e., P(F2|I).
### Application of Bayes' Theorem
Bayes' Theorem formula is:
\[ P(F2|I) = \frac{P(I|F2) \cdot P(F2)}{P(I)} \]
Where \( P(I) \) is the total probability of a form being processed incorrectly. This can be calculated by summing the probabilities of incorrect processing from both firms:
\[ P(I) = P(I|F1) \cdot P(F1) + P(I|F2) \cdot P(F2) \]
Substitute the given values:
\[ P(I) = (0.02 \cdot 0.80) + (0.05 \cdot 0.20) \]
\[ P(I) = 0.016 + 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F464fe16a-00d5-4182-8379-61bd4b8bbe02%2F31b6d25d-daf2-41b3-8a25-5370387ec046%2F3qe3okp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Probability and Data Processing: A Case Study of Subcontracted Work
**Problem Statement:**
An insurance company subcontracts two data processing firms to handle their forms. Firm #1 handles 80% of the forms, while Firm #2 handles the rest. Unfortunately, these firms are run by people and are therefore not perfect. Here are some key details:
- It is known that 2% of all forms processed by Firm #1 will be processed incorrectly.
- 5% of the forms from Firm #2 will also be processed incorrectly.
**Question:**
Suppose a random form is chosen from those that have been processed. What is the probability that this form was processed by Firm #2, given that it was processed incorrectly? (Hint: use a probability tree or Bayes' Theorem for the solution.)
**Detailed Explanation:**
When solving this problem, it's helpful to use Bayes' Theorem, which helps us revise existing predictions based on new evidence. Also, we can use a probability tree to visualize the problem.
### Information Recap
- Probability a form is handled by Firm #1: P(F1) = 0.80
- Probability a form is handled by Firm #2: P(F2) = 0.20
- Probability of incorrect processing by Firm #1: P(I|F1) = 0.02
- Probability of incorrect processing by Firm #2: P(I|F2) = 0.05
Our task is to find the probability that a form was handled by Firm #2 given that it was processed incorrectly, i.e., P(F2|I).
### Application of Bayes' Theorem
Bayes' Theorem formula is:
\[ P(F2|I) = \frac{P(I|F2) \cdot P(F2)}{P(I)} \]
Where \( P(I) \) is the total probability of a form being processed incorrectly. This can be calculated by summing the probabilities of incorrect processing from both firms:
\[ P(I) = P(I|F1) \cdot P(F1) + P(I|F2) \cdot P(F2) \]
Substitute the given values:
\[ P(I) = (0.02 \cdot 0.80) + (0.05 \cdot 0.20) \]
\[ P(I) = 0.016 + 0.
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