Probability of Hospital Preference Given Positive Test

In a certain town, there are two hospitals, Hospital A and Hospital B. It is known that 70% of the patients with a specific disease prefer Hospital A, while 30% prefer Hospital B. The disease has a prevalence of 1% in the total population. A diagnostic test for the disease has been developed that is 90% accurate, meaning that it correctly identifies the disease 90% of the time in those who have it (true positive rate) and correctly identifies that a person does not have the disease 90% of the time in those who do not have it (true negative rate).

You randomly select a patient from the general population who has tested positive for the disease. What is the probability that this patient prefers Hospital A?

Use Bayes' Theorem to solve this problem. Let:

- $P(A)$ be the probability that a patient prefers Hospital A.

- $P(B)$ be the probability that a patient prefers Hospital B.

- $P(D|A)$ be the probability of testing positive given the preference for Hospital A.

- $P(D|B)$ be the probability of testing positive given the preference for Hospital B.

First, find the necessary probabilities and apply Bayes' Theorem: $$P(A|D) = \frac{P(D|A)P(A)}{P(D)}$$