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To solve this problem using Bayes' Theorem, let’s break down the information given and apply the theorem accordingly.

### Given Data:
1. **Probability of being infected, P(A)**: \( \frac{1}{150} \approx 0.00667 \)
2. **Probability of testing positive given infection, P(B|A)**: 70% or 0.70
3. **Probability of testing positive given no infection (false positive), P(B|A')**: 15% or 0.15

We are asked to find:
- **(a) The probability that a person is infected given a positive test result, P(A|B)**
- **(b) The probability that a person is not infected given a negative test result, P(A'|B')**

### Solution

#### (a) Using Bayes' Theorem, we calculate \( P(A|B) \):
Bayes' Theorem states:
\[
P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}
\]
where
\[
P(B) = P(B|A) \cdot P(A) + P(B|A') \cdot P(A')
\]
1. **Calculate \( P(A') \)**: This is the probability of not being infected.
   \[
   P(A') = 1 - P(A) = 1 - 0.00667 = 0.99333
   \]
2. **Calculate \( P(B) \)**:
   \[
   P(B) = (0.70 \cdot 0.00667) + (0.15 \cdot 0.99333)
   \]
   \[
   P(B) = 0.004669 + 0.149
   \]
   \[
   P(B) = 0.153669
   \]
3. **Calculate \( P(A|B) \)**:
   \[
   P(A|B) = \frac{0.70 \cdot 0.00667}{0.153669}
   \]
   \[
   P(A|B) \approx \frac{0.004669}{0.153669} \approx 0.0304
   \]

So, **the probability that a person is infected given a positive test result is approximately 0.030**.

#### (b) Using Bayes' Theorem, we calculate \( P(A'|B') \):
Using Bayes' Theorem, this becomes:
\[
P(A'|B') = \frac{P(B'|A') \cdot P(A')}{P(B')}
\]
where
\[
P(B') = P(B'|A) \cdot P(A) + P(B'|A') \cdot P(A')
\]
We already know:
- \( P(B'|A) = 1 - P(B|A) = 1 - 0.70 = 0.30 \)
- \( P(B'|A') = 1 - P(B|A') = 1 - 0.15 = 0.85 \)

1. **Calculate \( P(B') \)**:
   \[
   P(B') = (0.30 \cdot 0.00667) + (0.85 \cdot 0.99333)
   \]
   \[
   P(B') = 0.002001 + 0.844331
   \]
   \[
   P(B') = 0.846332
   \]
2. **Calculate \( P(A'|B') \)**:
   \[
   P(A'|B') = \frac{0.85 \cdot 0.99333}{0.846332}
   \]
   \[
   P(A'|B') \approx \frac{0.844331}{0.846332} \approx 0.9976
   \]

So, **the probability that a person is not infected given a negative test result is approximately 0.998**.

Would you like further details on any steps?

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### Follow-up Questions:
1. What is Bayes' Theorem, and why is it useful in medical testing?
2. How does the false positive rate affect the probability of actually having the virus after a positive test?
3. What happens to \( P(A|B) \) as the infection rate increases?
4. How would a higher sensitivity of the test affect these probabilities?
5. Can we use Bayes' Theorem if multiple conditions or diseases are involved?

#### Tip:
Always verify given probabilities and the total probability calculation, as small errors in probabilities can lead to large errors in results.

A certain virus infects one in every 150 people. A test used to detect the virus in a person is positive 70% of the time when the person has the virus and 15% of the time when the person does not have the virus (false positive). Using Bayes' Theorem:

(a) When a person tests positive, determine the probability that the person is infected.
(b) When a person tests negative, determine the probability that the person is not infected.

Learn how to calculate the probability of infection based on positive and negative test results using Bayes' Theorem. This problem covers essential concepts in probability and conditional probability, using a real-life example involving virus testing accuracy.

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