Sample spaces, complements and mutually exclusive events

Core ideas

Sample spaces and equally likely outcomes

A random process is a repeatable situation whose outcome is unknown in advance, but whose long-run pattern of outcomes is predictable.

• The sample space ($S$) is the set of all possible non-overlapping outcomes of a random process.
• An event is any collection (subset) of outcomes from the sample space.

Every valid probability model obeys two foundational rules:

1. For any event $A$, $0 \le P(A) \le 1$.
2. The sum of the probabilities of all outcomes in the sample space equals $1$: $P(S) = 1$.

When all outcomes in a finite sample space $S$ are equally likely, the probability of an event $A$ is:

The complement rule

The complement of event $A$, denoted $A^C$ (or $A'$), consists of all outcomes in the sample space that are not in $A$. Because an event either occurs or does not occur:

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