Bayes' Theorem Application in Medical Diagnosis

In a certain medical test for a rare disease, the probability of a person having the disease given a positive test result is determined using Bayes' Theorem. The following information is known:

• The probability of having the disease (prior probability), $P(D) = 0.01$.
• The probability of a positive test result given that the person has the disease (sensitivity), $P(T^+ | D) = 0.95$.
• The probability of a positive test result given that the person does not have the disease (false positive rate), $P(T^+ | D^c) = 0.10$.

What is the probability that a person actually has the disease, given that they received a positive test result?

Use Bayes' Theorem, which states:

$$P(D | T^+) = \f\frac{P(T^+ | D) \cdot P(D)}{P(T^+)}$$

where $P(T^+)$ can be calculated as:

$$P(T^+) = P(T^+ | D) \cdot P(D) + P(T^+ | D^c) \cdot P(D^c)$$