Apply Bayes' rule to compute posterior

A binary event $A$ (for example, “user is a payer” or “patient has a disease”) occurs with prior probability $P(A)$. You observe evidence $B$ (such as a model predicting positive, or a test result coming back positive) with known likelihoods:

• $P(B\mid A)$
• $P(B\mid\neg A)$ Task:

1. Use Bayes’ rule to compute the posterior probability $P(A\mid B)$.
2. Show how to derive $P(B)$ from the given quantities. Optionally, include a numeric example to check the formula. Overview: Tests understanding of Bayes’ rule and basic probability concepts—specifically computing a posterior probability $P(A\mid B)$ and the marginal likelihood $P(B)$ from a prior and likelihoods—in the Statistics & Math domain at a foundational-to-intermediate abstraction level.