1.2 Mutually Exclusive Events

Mutually Exclusive Events

If events $A_1,...,A_N\subseteq \Omega$ in the same probability space are disjoint, such that $A_i\cap A_j=\varnothing$ for all $i\ne j$ (i.e. the events do not share common sample points), then by the Law of Finite Additivity, the probability that one of the events occurs is equal to the sum of their probabilities,
$\mathbb{P}[A_1\cup \dots \cup A_N]=P[A_1]+\dots +P[A_N].$
Such events are called mutually exclusive, and cannot occur simultaneously.

Examples

Dice Roll

A 6-sided dice may have many events. Two of these events are $A=\{2,4,6\}$ (i.e. rolling even), and $B=\{1,3,5\}$ (i.e. rolling odd). Their intersection is $A\cap B=\varnothing$. Therefore, these events are mutually exclusive; you cannot roll odd and even at the same time.

However, consider the event $C=\{3,6\}$ (rolling a multiple of 3). Then, $A\cap C=\{6\}$, and $B\cap C=\{3\}$. Therefore, neither $A$ nor $B$ are mutually exclusive to $C$, since you can both roll a multiple of 3 and an even number, or roll a multiple of 3 and an odd number.