how to prove Bayes theorem?

Question

how to prove Bayes theorem? (Develop from conditional probability ) 2 criterias -Total of law is probability -multiplication rule Can someone show me the mathematical proof & steps for this please?

Answer 1

Hints

• Write $P(A \cap B)$ two different ways using conditional probability
• these are therefore equal by Euclid's first axiom
• manipulate into Bayes' theorem (assuming none of the probabilities are $0$)

Answer 2

The conditional probability is defined as $$ P(A|B) = \frac{P(A\cap B)}{P(B)}, $$ hence $$ P(A|B) P(B) = P(B|A) P(A) . $$