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The problem asks to compute the probability that a person who tests positive for a disease actually has it, given the following information:

- The incidence rate of the disease is 0.5%.
- The false negative rate is 5%.
- The false positive rate is 3%.

This is a classic application of **Bayes' Theorem**, which calculates conditional probabilities. The formula for Bayes' Theorem is:

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

Where:
- \( P(A|B) \) is the probability that the person actually has the disease given they tested positive.
- \( P(A) \) is the prior probability that a person has the disease (the incidence rate).
- \( P(B|A) \) is the probability of testing positive given that the person has the disease (1 - false negative rate).
- \( P(B|\neg A) \) is the probability of testing positive given that the person does not have the disease (false positive rate).
- \( P(\neg A) \) is the probability that a person does not have the disease (1 - incidence rate).

Let's define the values:
- \( P(A) = 0.005 \) (incidence rate of 0.5%).
- \( P(B|A) = 0.95 \) (since the false negative rate is 5%, the probability of a true positive is 95%).
- \( P(B|\neg A) = 0.03 \) (false positive rate of 3%).
- \( P(\neg A) = 0.995 \) (since 1 - 0.005 = 0.995, the probability of not having the disease).

Now we can compute the probability \( P(A|B) \).

\[
P(A|B) = \frac{0.95 \times 0.005}{(0.95 \times 0.005) + (0.03 \times 0.995)}
\]

Let me calculate this.The probability that a person who tests positive actually has the disease is approximately **0.137** or **13.7%**, when rounded to three decimal places.

Do you have any questions or would you like further clarification?

Here are five related questions to expand on this:

1. How would the probability change if the incidence rate of the disease were higher?
2. What happens to the probability if the false positive rate increases?
3. Can you calculate the probability that a person who tests negative does not have the disease?
4. How would the results change if the test sensitivity (true positive rate) were lower?
5. What real-world applications might involve Bayes' Theorem beyond medical testing?

**Tip**: When working with Bayes' Theorem, always ensure that you correctly identify the prior probability and the likelihoods to avoid common errors in conditional probability calculation.

A certain disease has an incidence rate of 0.5%. If the false negative rate is 5% and the false positive rate is 3%, compute the probability that a person who tests positive actually has the disease.

Learn how to apply Bayes' Theorem to calculate the probability of having a disease given a positive test result. The problem involves a disease with an incidence rate of 0.5%, a false negative rate of 5%, and a false positive rate of 3%. The solution provides a detailed calculation using conditional probability. Suitable for high school students or introductory college-level probability courses.

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