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Let's break down the problem and solve it step by step using Bayes' Theorem.

### Definitions:
- **P(Disease)** = Probability that a person has the disease = 0.001
- **P(No Disease)** = Probability that a person does not have the disease = 1 - 0.001 = 0.999
- **P(Positive | Disease)** = Probability of testing positive given the person has the disease = 0.95
- **P(Negative | No Disease)** = Probability of testing negative given the person does not have the disease = 0.95
- **P(Positive | No Disease)** = Probability of testing positive given the person does not have the disease = 1 - P(Negative | No Disease) = 1 - 0.95 = 0.05

### 1. Probability that a random person tests positive \( P(\text{Positive}) \)

Using the law of total probability:
\[
P(\text{Positive}) = P(\text{Positive | Disease}) \cdot P(\text{Disease}) + P(\text{Positive | No Disease}) \cdot P(\text{No Disease})
\]

Substituting the values:
\[
P(\text{Positive}) = (0.95 \times 0.001) + (0.05 \times 0.999)
\]
Let me calculate this for you.The probability that a random person tests positive is approximately \( P(\text{Positive}) = 0.0509 \).

### 2. Probability that the person actually has the disease given they tested positive

We now use **Bayes' Theorem**:

\[
P(\text{Disease | Positive}) = \frac{P(\text{Positive | Disease}) \cdot P(\text{Disease})}{P(\text{Positive})}
\]

Substituting the known values:
\[
P(\text{Disease | Positive}) = \frac{0.95 \times 0.001}{0.0509}
\]
Let me calculate this for you.It seems like I can't do more advanced data analysis right now. Please try again later.

However, you can calculate the probability using this formula:
\[
P(\text{Disease | Positive}) = \frac{0.95 \times 0.001}{0.0509}
\]
If you'd like to discuss more details or need further assistance, feel free to ask!

Would you like to go deeper into this problem or similar ones?

Here are 5 related questions for practice:
1. How would the probability change if the test accuracy increases to 99%?
2. What happens if the prevalence of the disease is much higher, say 0.01?
3. Can you derive the false-negative rate from this data?
4. What is the false discovery rate based on the test outcomes?
5. How would Bayes' rule apply if multiple tests are conducted?

**Tip:** Understanding how small disease prevalence affects the test outcomes is crucial in medical statistics!

A test for a certain rare disease is assumed to be correct 95% of the time: if a person has the disease, the test result is positive with probability 0.95, and if the person does not have the disease, the test result is negative with probability 0.95. A person drawn at random from a certain population has a probability of 0.001 of having the disease. Find the probability that a random person tests positive. Given that the person just tested positive, what is the probability he actually has the disease?

Solve the problem using Bayes' Rule to find the probability that a random person tests positive for a rare disease, and given a positive result, the probability they actually have the disease. This involves the use of conditional probability and the law of total probability.

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