Proof: Monotonicity of probability
Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Probability axioms ▷ Monotonicity of probability
Metadata: ID: P243 | shortcut: prob-mon | author: JoramSoch | date: 2021-07-30, 11:37.
Theorem: Probability is monotonic, i.e. if $A$ is a subset of or equal to $B$, then the probability of $A$ is smaller than or equal to $B$:
\[\label{eq:prob-mon} A \subseteq B \quad \Rightarrow \quad P(A) \leq P(B) \; .\]Proof: Set $E_1 = A$, $E_2 = B \setminus A$ and $E_i = \emptyset$ for $i \geq 3$. Then, the sets $E_i$ are pairwise disjoint and $E_1 \cup E_2 \cup \ldots = B$, because $A \subseteq B$. Thus, from the third axiom of probability, we have:
\[\label{eq:pB} P(B) = P(A) + P(B \setminus A) + \sum_{i=3}^\infty P(E_i) \; .\]Since, by the first axiom of probability, the right-hand side is a series of non-negative numbers converging to $P(B)$ on the left-hand side, it follows that
\[\label{eq:prob-mon-qed} P(A) \leq P(B) \; .\]∎
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#Monotonicity.
Metadata: ID: P243 | shortcut: prob-mon | author: JoramSoch | date: 2021-07-30, 11:37.