Proof

(related to Proposition: Characterization of Independent Events II)

The proposition follows immediately from the other characterization of independent events, according to which given an event \(B\) with \(0 < p(B) < 1\), the event \(A\) is independent from \(B\) if and only if

\[p(A)=p(A|B).\]

By definition of conditional probability, it follows

\[p(A)=p(A|B)=\frac{p(A\cap B)}{p(B)}, \] which is equivalent to \[p(A)\cdot p(B)=p(A\cap B).\]

Assuming that \(A\) is independent from \(B\), like shown above, and in addition assuming \(0 < p(A) < 1\), we can conclude that \(B\) is independent from \(A\), for symmetry reasons.


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References

Bibliography

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