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?

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.