For any two events \(A\) and \(B\), the probability of the joint event \(A\cap B\) (i.e. the events \(A\) and \(B\) happening concurrently), can be calculated by the product of the conditional probability \(p(A|B)\) and the probability of \(B\). Alternatively, we can calculate it as the product of the conditional probability \(p(B|A)\) and the probability of \(A\). Altogether, the following formulas hold:
\[p(A\cap B)= p(A|B) \cdot p(B)=p(B|A)\cdot p(A).\]
Proofs: 1
Definitions: 1 2
Proofs: 3 4
Propositions: 5