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The problem is asking for the probability that a person actually has a disease given that they tested positive for it. This is a classic example of applying **Bayes' Theorem**. Let's define the events:

- \( A \): The person has the disease.
- \( B \): The test result is positive for the disease.

We are given:
- \( P(B | A) = 0.95 \): The probability of testing positive given that a person has the disease.
- \( P(B | A^c) = 0.001 \): The probability of testing positive given that a person does not have the disease.
- \( P(A) = 0.0017 \): The proportion of people in the population who have the disease.

We want to find \( P(A | B) \), the probability that a person has the disease given that they tested positive.

### Applying Bayes' Theorem

Bayes' Theorem is given by:

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

Where \( P(B) \) is the total probability of a positive test result, which can be found using the law of total probability:

\[
P(B) = P(B | A) \cdot P(A) + P(B | A^c) \cdot P(A^c)
\]

Now let's compute the values:

\[
P(A^c) = 1 - P(A) = 1 - 0.0017 = 0.9983
\]

\[
P(B) = (0.95 \times 0.0017) + (0.001 \times 0.9983)
\]

### Calculating \( P(B) \)

Let's compute this step-by-step.The total probability of a positive test result, \( P(B) \), is approximately \( 0.0026133 \).

### Calculating \( P(A | B) \)

Now we use Bayes' Theorem to calculate \( P(A | B) \):

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

Substitute the values:

\[
P(A | B) = \frac{0.95 \times 0.0017}{0.0026133}
\]

Let's compute this.It seems like I can’t do more advanced data analysis right now. Please try again later. 

However, I can guide you through the final calculation manually:

### Manual Calculation

To calculate \( P(A | B) \):

\[
P(A | B) = \frac{0.95 \times 0.0017}{0.0026133}
\]

1. Multiply \( 0.95 \times 0.0017 = 0.001615 \).
2. Then divide by \( 0.0026133 \).

So, \( P(A | B) \approx \frac{0.001615}{0.0026133} \approx 0.618 \).

This means the probability that a person actually has the disease given a positive test result is approximately **61.8%**.

If you have more questions or want further details, feel free to ask!

### Related Questions
1. What is the significance of Bayes' Theorem in medical testing?
2. How does the accuracy of a test impact the post-test probability?
3. What are false positives and false negatives in the context of medical testing?
4. How can the sensitivity and specificity of a test be interpreted?
5. How does the prevalence of a disease affect the probability of actually having the disease after a positive test?

### Tip
Understanding the **base rate fallacy** can help in interpreting probabilities correctly, especially in medical diagnostics where rare conditions are involved.

Considere um teste de laboratório para detectar uma determinada doença. Definem-se os seguintes eventos:
A = pessoa tem a doença.
B = teste teve resultado positivo para a doença.
Sabe-se que:
P(B | A) = 0.95
P(B | A^c) = 0.001
e que a proporção de doentes da população é 0.0017. Qual a probabilidade de que uma pessoa tenha de fato a doença dado que o teste foi positivo?

Learn how to calculate the probability that a person has a disease given a positive test result using Bayes' Theorem. The example involves a test with sensitivity of 95%, false positive rate of 0.1%, and a disease prevalence of 0.17%.

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