Bayes' Theorem Application in Clinical Testing

A pharmaceutical company is conducting a clinical trial to test a new medication aimed at reducing the risk of heart disease. Based on prior studies, it is known that:

• The probability that an individual has heart disease is $P(H) = 0.1$.
• The probability that an individual tests positive for heart disease if they have the disease is $P(T^+ | H) = 0.9$.
• The probability that an individual tests positive for heart disease if they do not have the disease is $P(T^+ | H^c) = 0.2$.

If a patient tests positive for heart disease, what is the probability that they actually have the disease?

Apply Bayes' Theorem to calculate this probability, which can be expressed as:

$P(H | T^+) = \frac{P(T^+ | H) imes P(H)}{P(T^+)}$

where $P(T^+)$ can be derived using the law of total probability.