Index: The Book of Statistical ProofsGeneral Theorems ▷ Probability theory ▷ Probability axioms ▷ Range of probability

Theorem: The probability of an event is bounded between 0 and 1:

$\label{eq:prob-range} 0 \leq P(E) \leq 1 \; .$

Proof: From the first axiom of probability, we have:

$\label{eq:pEg0} P(E) \geq 0 \; .$

By combining the first axiom of probability and the probability of the complement, we obtain:

$\label{eq:pEl1} \begin{split} 1- P(E) = P(E^\mathrm{c}) &\geq 0 \\ 1- P(E) &\geq 0 \\ P(E) &\leq 1 \; . \end{split}$

Together, \eqref{eq:pEg0} and \eqref{eq:pEl1} imply that

$\label{eq:prob-range-qed} 0 \leq P(E) \leq 1 \; .$
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Metadata: ID: P246 | shortcut: prob-range | author: JoramSoch | date: 2021-07-30, 12:25.