Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Probability axioms ▷ Boole's inequality

Theorem: The probability of any countable sequence of events $A_1, A_2, A_3, \ldots$ is smaller than or equal to the sum of the probabilities of the individual events:

\[\label{eq:bool-ineq} P\left( \bigcup_{i=1}^\infty A_i \right) \leq \sum_{i=1}^\infty P(A_i) \; .\]

Proof: The addition law of probability states that, for two events $A$ and $B$, we have:

\[\label{eq:prob-add} P(A \cup B) = P(A) + P(B) - P(A \cap B) \; .\]

We will prove the statement by induction, i.e. observe that it holds for $n=1$ event and show that, if it holds for $n$ events, it also holds for $n+1$ events. For $n=1$, it is obviously true that:

\[\label{eq:bi-1} P\left( A_1 \right) \leq P(A_1) \; .\]

We shall suppose that statement \eqref{eq:bool-ineq} holds for $A_1, \ldots, A_n$:

\[\label{eq:bi-n} P\left( \bigcup_{i=1}^n A_i \right) \leq \sum_{i=1}^n P(A_i) \; .\]

Using \eqref{eq:prob-add}, for $A_1, \ldots, A_n, A_{n+1}$, we then have:

\[\label{eq:bi-n+1-s1} P\left( \bigcup_{i=1}^{n+1} A_i \right) = P\left( \bigcup_{i=1}^{n} A_i \cup A_{n+1} \right) = P\left( \bigcup_{i=1}^{n} A_i \right) + P(A_{n+1}) - P\left( \bigcup_{i=1}^{n} A_i \cap A_{n+1} \right) \; .\]

Since, by the first axiom of probability, any probability is positive

\[\label{eq:prob-pos} \begin{split} P\left( \bigcup_{i=1}^{n} A_i \cap A_{n+1} \right) \geq 0 \; , \end{split}\]

it follows from \eqref{eq:bi-n+1-s1} that

\[\label{eq:bi-n+1-s2} P\left( \bigcup_{i=1}^{n+1} A_i \right) \leq P\left( \bigcup_{i=1}^{n} A_i \right) + P(A_{n+1})\]

and using \eqref{eq:bi-n}, we get

\[\label{eq:bi-n+1-s3} P\left( \bigcup_{i=1}^{n+1} A_i \right) \leq \sum_{i=1}^n P(A_i) + P(A_{n+1}) \; ,\]

such that

\[\label{eq:bi-n+1-s4} P\left( \bigcup_{i=1}^{n+1} A_i \right) \leq \sum_{i=1}^{n+1} P(A_i) \; .\]

Together, \eqref{eq:bi-1} and the transition from \eqref{eq:bi-n} to \eqref{eq:bi-n+1-s4}, prove the theorem.

∎
Sources:

Metadata: ID: P483 | shortcut: bool-ineq | author: JoramSoch | date: 2025-01-10, 13:10.