Range of probability

Index: The Book of Statistical Proofs ▷ General 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:

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

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

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

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

∎

Sources:

A.N. Kolmogorov (1950): "Elementary Theory of Probability"; in: Foundations of the Theory of Probability, p. 6; URL: https://archive.org/details/foundationsofthe00kolm/page/6/mode/2up.
Alan Stuart & J. Keith Ord (1994): "Probability and Statistical Inference"; in: Kendall's Advanced Theory of Statistics, Vol. 1: Distribution Theory, pp. 288-289; URL: https://www.wiley.com/en-us/Kendall%27s+Advanced+Theory+of+Statistics%2C+3+Volumes%2C+Set%2C+6th+Edition-p-9780470669549.
Wikipedia (2021): "Probability axioms"; in: Wikipedia, the free encyclopedia, retrieved on 2021-07-30; URL: https://en.wikipedia.org/wiki/Probability_axioms#The_numeric_bound.

Metadata: ID: P246 | shortcut: prob-range | author: JoramSoch | date: 2021-07-30, 12:25.