Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Probability axioms ▷ Monotonicity of probability

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

\[\label{eq:prob-mon-qed} P(A) \leq P(B) \; .\]
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Metadata: ID: P243 | shortcut: prob-mon | author: JoramSoch | date: 2021-07-30, 11:37.