Proposition: Probability of Joint Events

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


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References

Bibliography

  1. Bosch, Karl: "Elementare Einf├╝hrung in die Wahrscheinlichkeitsrechnung", vieweg Studium, 1995, 6th Edition