Explanation: Mutually Exclusive vs. Pairwise Independent Events

(related to Definition: Pairwise Independent Events)

Mutually exclusive events cannot be pairwise independent, i.e. from

\[A_i\cap A_j=\emptyset\]

it does not (!) follow

\[0=p(A_i\cap A_j)=p(A_i)\cdot p(A_j),\quad\quad( * )\]

for all

\[A_i,A_j\subseteq\Omega, i\neq j.\]

In particular, at least one of the probabilities in would have to be \(0\) to fulfill \( ( * ) \), i.e. it would be the impossible event. In other words, e.g. if we assume \(p(A_i) > 0\), then the event \(A_i\) can occur. But then \(p(A_j)\) drops to \(0\). Thus, \(A_j\) cannot occur, if \(A_i\) can, and vice versa. Therefore, both events are not independent (i.e. their occurrence influences each other).


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References

Bibliography

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