Proof of Bayes' Theorem from First Principles of Conditional Probability

The 'Base Rate Fallacy' is the primary trap. Students frequently mistake the likelihood

P(B|A)

for the posterior

P(A|B)

. Without incorporating the prior

P(A)

, one might assume a positive medical test implies a high disease probability, even when the disease itself is vanishingly rare.

Institutional Reference: Applied Statistics