This is a classic case of conditional probability, and we can use Bayes' Theorem to solve it.

Given:

• 1% of the population has the disease $D$ (i.e., $P(D) = 0.01$).
• The probability that the test is positive given the person has the disease is $P(\text{Positive} | D) = 0.95$ (true positive rate).
• The probability of a false negative (test is negative given the person has the disease) is 5%, which is $P(\text{Negative} | D) = 0.05$.
• The probability that the test is negative given the person does not have the disease is $P(\text{Negative} | \neg D) = 0.98$.
• The probability of a false positive (test is positive given the person does not have the disease) is 2%, which is $P(\text{Positive} | \neg D) = 0.02$.

We are asked to find:

$P(D | \text{Positive})$, the probability that a person has the disease given that the test result is positive.

Bayes' Theorem:

$P(D | \text{Positive}) = \frac{P(\text{Positive} | D) \cdot P(D)}{P(\text{Positive})}$

Where: $P(\text{Positive}) = P(\text{Positive} | D) \cdot P(D) + P(\text{Positive} | \neg D) \cdot P(\neg D)$ $P(\neg D) = 1 - P(D) = 0.99$.

Now, let's calculate:

1. $P(\text{Positive}) = (0.95 \cdot 0.01) + (0.02 \cdot 0.99)$ $P(\text{Positive}) = 0.0095 + 0.0198 = 0.0293$

2. Now, apply Bayes' Theorem: $P(D | \text{Positive}) = \frac{0.95 \cdot 0.01}{0.0293} = \frac{0.0095}{0.0293} \approx 0.324$

Answer:

The probability that the person actually has the disease given a positive test result is approximately 0.324 (or 32.4%).

Would you like any further details or explanations?

Here are 5 related questions you might be interested in:

1. How does increasing the disease prevalence affect the result?
2. What happens to the probability if the test's false positive rate decreases?
3. How does Bayes' Theorem apply in other medical contexts?
4. Can we calculate the probability of not having the disease given a negative test?
5. How does sample size affect confidence in medical tests?

Tip: Always be cautious with test results, especially when the disease prevalence is low, as false positives can be common!