Events and Their Types

An event is any subset of the sample space $S$ — that is, a collection of one or more outcomes that we are interested in. Events are usually written with capital letters $A, B, C, \ldots$. An event is said to occur when the actual outcome of the experiment is one of the outcomes belonging to that event.

For example, in rolling a die ($S = \{1,2,3,4,5,6\}$), "getting an even number" is the event $A = \{2, 4, 6\}$, and "getting a number greater than $4$" is $B = \{5, 6\}$.

Because events are sets, we combine them with set operations, each with a plain-language meaning:

• Union $A \cup B$ — the event "$A$ or $B$ (or both) occurs": all outcomes in $A$, in $B$, or in both.
• Intersection $A \cap B$ — the event "$A$ and $B$ both occur": outcomes common to both.
• Complement $A'$ (also $\bar{A}$ or $A^{c}$) — the event "$A$ does not occur": all outcomes of $S$ that are not in $A$.

Two special relationships between events are central to the theorems that follow:

• Events $A$ and $B$ are mutually exclusive (disjoint) if they cannot happen together — they share no outcome, i.e. $A \cap B = \varnothing$. Getting a head and getting a tail on one toss are mutually exclusive. …