Definition: Mutually Exclusive and Collectively Exhaustive Events

The events \(A_i\subseteq \Omega\) are said to be mutually exclusive, if any two of them cannot occur at once, i.e. if \(A_i\cap A_j\) is the impossible event, formally \[A_i\cap A_j=\emptyset,~~~~~~~~~~( A_i,A_j\subseteq\Omega, i\neq j).\]

The events \(A_1, A_2, \ldots, A_n\) are said to be collectively exhaustive, if at least one of them must occur, i.e. if the occurrence of any of these events is the certain event, formally

\[\bigcup_{i\in I} A_i=\Omega~~~~~~~~~~(I\text{ being a set of indices of all events in }\Omega).\]

In an experiment with mutually exclusive and collectively exhaustive events, exactly one event \(A_i\) will occur.

Axioms: 1
Definitions: 2 3
Explanations: 4
Proofs: 5 6 7 8 9 10 11 12 13
Propositions: 14
Solutions: 15
Theorems: 16 17

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