A certain disease has an incidence rate of 0.2%. If the false negative rate is 8% and the false positive rate is 4%, compute the probability that a person who tests positive actually has the disease.

This problem involves calculating the probability that a person who tests positive for a disease actually has the disease. This is a classic example of applying **Bayes' Theorem**.

Let's define the following:
- \( P(D) \) = Probability that a person has the disease = 0.2% = 0.002
- \( P(\neg D) \) = Probability that a person does not have the disease = 1 - 0.002 = 0.998
- \( P(\text{Pos} | D) \) = Probability of testing positive given the person has the disease = 1 - false negative rate = 1 - 0.08 = 0.92
- \( P(\text{Pos} | \neg D) \) = Probability of testing positive given the person does not have the disease = false positive rate = 0.04
- \( P(\text{Pos}) \) = Total probability of testing positive (to be computed).

We are asked to find the **probability that a person who tests positive actually has the disease**, which is:
\[
P(D | \text{Pos}) = \frac{P(\text{Pos} | D) P(D)}{P(\text{Pos})}
\]

To calculate \( P(\text{Pos}) \), we use the law of total probability:
\[
P(\text{Pos}) = P(\text{Pos} | D) P(D) + P(\text{Pos} | \neg D) P(\neg D)
\]
Substituting the known values:
\[
P(\text{Pos}) = (0.92 \times 0.002) + (0.04 \times 0.998)
\]
\[
P(\text{Pos}) = 0.00184 + 0.03992 = 0.04176
\]

Now, using Bayes' Theorem:
\[
P(D | \text{Pos}) = \frac{0.92 \times 0.002}{0.04176} = \frac{0.00184}{0.04176} \approx 0.04406
\]

Thus, the probability that a person who tests positive actually has the disease is approximately **4.41%**.

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Would you like further details on this process, or have any questions?

Here are 5 related questions you might explore:

1. How does increasing the false negative rate affect the probability of having the disease given a positive test result?
2. How does increasing the incidence rate of the disease change the outcome?
3. What would the probability be if the false positive rate were reduced to 2%?
4. How can Bayes' Theorem be applied to non-medical scenarios, such as spam detection?
5. What is the impact of testing sensitivity and specificity on disease detection rates?

**Tip:** When dealing with medical tests, remember that the incidence rate (or prior probability) heavily influences the probability of having a condition given a positive test result.

Learn how to use Bayes' Theorem to compute the probability that a person who tests positive for a disease actually has it. This example, suitable for students in Grades 10-12, involves conditional probability, false positive and false negative rates, and incidence rates.

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