This problem can be solved using Bayes' Theorem, which calculates the probability of an event given the occurrence of another event. Here, we are asked to find the probability that a person has the disease given that the test result is positive.

Known Information:

• $P(\text{Positive Test} \mid \text{Disease}) = 0.95$ (Test detects disease correctly 95% of the time)
• $P(\text{Positive Test} \mid \text{No Disease}) = 0.01$ (False positive rate of 1%)
• $P(\text{Disease}) = 0.005$ (0.5% of the population has the disease)
• $P(\text{No Disease}) = 0.995$ (99.5% of the population does not have the disease)

We need to compute: $P(\text{Disease} \mid \text{Positive Test})$

According to Bayes' Theorem:

$P(\text{Disease} \mid \text{Positive Test}) = \frac{P(\text{Positive Test} \mid \text{Disease}) \cdot P(\text{Disease})}{P(\text{Positive Test})}$

Where $P(\text{Positive Test})$ can be calculated as: $P(\text{Positive Test}) = P(\text{Positive Test} \mid \text{Disease}) \cdot P(\text{Disease}) + P(\text{Positive Test} \mid \text{No Disease}) \cdot P(\text{No Disease})$

Step-by-Step Table Representation:

Case	Probability of the Case (Prior)	Test Result Probability (Conditional)	Joint Probability
Has Disease	$P(\text{Disease}) = 0.005$	$P(\text{Positive} \mid \text{Disease}) = 0.95$	$0.005 \times 0.95 = 0.00475$
No Disease (False Positive)	$P(\text{No Disease}) = 0.995$	$P(\text{Positive} \mid \text{No Disease}) = 0.01$	$0.995 \times 0.01 = 0.00995$

The total probability of a positive test result is the sum of the joint probabilities: $P(\text{Positive Test}) = 0.00475 + 0.00995 = 0.0147$

Now applying Bayes' Theorem:

$P(\text{Disease} \mid \text{Positive Test}) = \frac{0.00475}{0.0147} \approx 0.323$

Final Answer:

The probability that a person has the disease given that the test result is positive is approximately 32.3%.

Would you like further details on any of the steps, or do you have any questions?

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Here are five related questions you could explore:

1. How would the result change if the population had a higher prevalence of the disease?
2. What happens to the probability if the false positive rate increases?
3. Can you calculate the probability that a person does not have the disease given a negative test?
4. How does the effectiveness of a test influence the outcome of Bayes' Theorem?
5. What other real-world situations can Bayes' Theorem be applied to?

Tip: When dealing with probabilities, always identify the relevant prior probabilities and conditional probabilities before applying Bayes' Theorem.