A certain virus infects one in every 200 people. A test used to detect the virus in a person is positive 90% of the time if the person has the virus and 8% of the time if the person does not have the virus. (This 8% result is called a false positive.) Let A be the event "the person is infected" and B be the event "the person tests positive".

a) Find the probability that a person has the virus given that they have tested positive, i.e. find P(A|B). Round your answer to a percent rounded to the nearest fourth decimal place and do not include a percent sign.
P(A|B)= 
%

b) Find the probability that a person does not have the virus given that they test negative, i.e. find P(A'|B'). Round your answer as a percent rounded to the nearest fourth decimal place and do not include a percent sign.
P(Ā|B̄) = 
%


We can solve both parts of this problem using **Bayes' Theorem** and the **law of total probability**. 

### Given information:
- The probability a person is infected:  
  \( P(A) = \frac{1}{200} = 0.005 \)
  
- The probability a person is not infected:  
  \( P(A') = 1 - P(A) = 0.995 \)

- The probability of testing positive given the person has the virus (true positive):  
  \( P(B|A) = 0.90 \)

- The probability of testing positive given the person does not have the virus (false positive):  
  \( P(B|A') = 0.08 \)

### a) Find \( P(A|B) \):
We want the probability that a person has the virus given that they tested positive. This is written as \( P(A|B) \), and it can be calculated using Bayes' Theorem:

\[
P(A|B) = \frac{P(B|A)P(A)}{P(B)}
\]

Now, we need to calculate \( P(B) \), the total probability that the test is positive, which can be found using the law of total probability:

\[
P(B) = P(B|A)P(A) + P(B|A')P(A')
\]

Substitute the known values:

\[
P(B) = (0.90)(0.005) + (0.08)(0.995)
\]

\[
P(B) = 0.0045 + 0.0796 = 0.0841
\]

Now, using Bayes' Theorem:

\[
P(A|B) = \frac{(0.90)(0.005)}{0.0841}
\]

\[
P(A|B) = \frac{0.0045}{0.0841} \approx 0.0535
\]

Therefore, the probability that a person has the virus given that they tested positive is approximately:

\[
P(A|B) \approx 0.0535
\]

### b) Find \( P(A'|B') \):
We want the probability that a person does **not** have the virus given that they tested negative, i.e. \( P(A'|B') \). Again, we use Bayes' Theorem:

\[
P(A'|B') = \frac{P(B'|A')P(A')}{P(B')}
\]

First, we calculate \( P(B') \), the probability of testing negative. Using the law of total probability:

\[
P(B') = P(B'|A)P(A) + P(B'|A')P(A')
\]

Where:
- \( P(B'|A) = 1 - P(B|A) = 1 - 0.90 = 0.10 \)
- \( P(B'|A') = 1 - P(B|A') = 1 - 0.08 = 0.92 \)

Substitute the known values:

\[
P(B') = (0.10)(0.005) + (0.92)(0.995)
\]

\[
P(B') = 0.0005 + 0.9144 = 0.9149
\]

Now, using Bayes' Theorem:

\[
P(A'|B') = \frac{(0.92)(0.995)}{0.9149}
\]

\[
P(A'|B') = \frac{0.914}{0.9149} \approx 0.9990
\]

Therefore, the probability that a person does not have the virus given that they tested negative is approximately:

\[
P(A'|B') \approx 0.9990
\]

### Final answers:
- \( P(A|B) \approx 5.35 \% \)
- \( P(A'|B') \approx 99.90 \% \)

Would you like further clarification on any steps or details in the calculations?

### 5 Relative Questions:
1. How would the probabilities change if the accuracy of the test (true positive rate) increased?
2. What happens to \( P(A|B) \) if the false positive rate decreases?
3. Can you calculate \( P(B|A') \) for different levels of infection rates?
4. How does Bayes' Theorem apply to real-world diagnostics in other fields, like cancer testing?
5. What would be the effect of a different infection rate on the result?

### Tip:
Bayes' Theorem is powerful for medical diagnostics, helping assess how reliable a positive or negative test result really is based on test accuracy and overall infection rates.

A certain virus infects one in every 200 people. A test used to detect the virus in a person is positive 90% of the time if the person has the virus and 8% of the time if the person does not have the virus. (This 8% result is called a false positive.) Let A be the event 'the person is infected' and B be the event 'the person tests positive'.

a) Find the probability that a person has the virus given that they have tested positive, i.e., find P(A|B). Round your answer to a percent rounded to the nearest fourth decimal place and do not include a percent sign.
P(A|B) = %

b) Find the probability that a person does not have the virus given that they test negative, i.e., find P(A'|B'). Round your answer as a percent rounded to the nearest fourth decimal place and do not include a percent sign.
P(Ā|B̄) = %

Explore the use of Bayes' Theorem to calculate the probability of a person having a virus given a positive test result. This problem covers the law of total probability and conditional probability, suitable for Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation