Types of Events — Exhaustive and Mutually Exclusive Events

Types of Events — Exhaustive and Mutually Exclusive Events

Two properties of a collection of events matter enormously when their probabilities are combined: whether the events can occur simultaneously, and whether they cover every possible outcome between them.

Mutually Exclusive Events

Two events $A$ and $B$ are called mutually exclusive if they cannot occur at the same time, i.e. if $A \cap B = \phi$. Equivalently, no outcome of the experiment belongs to both $A$ and $B$.

For a die thrown once, $A = \{1, 3, 5\}$ (odd) and $B = \{2, 4, 6\}$ (even) are mutually exclusive, since no number is both odd and even: $A \cap B = \phi$. But $A = \{1,3,5\}$ (odd) and $C = \{1, 2\}$ (less than $3$) are not mutually exclusive, since $1 \in A \cap C$.

The idea extends to any number of events $E_1, E_2, \dots, E_n$: they are (pairwise) mutually exclusive if $E_i \cap E_j = \phi$ for every pair $i \ne j$ — no two of them can happen together.

Exhaustive Events

A collection of events $E_1, E_2, \dots, E_n$ is called exhaustive if their union is the whole sample space:

This means every possible outcome of the experiment belongs to at least one of the events — between them, they leave nothing out.

For a die thrown once, $A = \{1,2,3\}$ and $B = \{4,5,6\}$ are exhaustive, because $A \cup B = \{1,2,3,4,5,6\} = S$.

Mutually Exclusive AND Exhaustive Events

Very often, as with $A = \{1,2,3\}$ and $B = \{4,5,6\}$ above, a collection of events is both mutually exclusive and exhaustive simultaneously: $A \cap B = \phi$ and $A \cup B = S$. Such a collection partitions the sample space into non-overlapping pieces that together account for every outcome — this is exactly the situation Axiom 3 of probability (next section) is built to handle. …