Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Probability axioms ▷ Probability of the complement

Theorem: The probability of a complement of a set is one minus the probability of this set:

\[\label{eq:prob-comp} p(\overline{A}) = 1 - p(A)\]

where $\overline{A} = \Omega \setminus A$ and $\Omega$ is the sample space.

Proof: Since $A$ and $\overline{A}$ are mutually exclusive and $A \cup \overline{A} = \Omega$, the third axiom of probability implies:

\[\label{eq:pAAc} \begin{split} p(A \cup \overline{A}) &= p(A) + p(\overline{A}) \\ p(\Omega) &= p(A) + p(\overline{A}) \\ p(\overline{A}) &= p(\Omega) - p(A) \; . \end{split}\]

The second axiom of probability states that $p(\Omega) =1$, such that we obtain:

\[\label{eq:prob-comp-qed} p(\overline{A}) = 1 - p(A) \; .\]
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Metadata: ID: P245 | shortcut: prob-comp | author: JoramSoch | date: 2021-07-30, 12:14.