Prove Theorem 2.4. That is, prove that

\[P\left(\bigcup_{i=1}^{n} A_{i}ight) \leq \sum_{i=1}^{n} P\left(A_{i}ight)\]

The most direct approach is based on mathematical induction using the general addition rule to prove the basis and the induction step. The general addition rule states that for any two events \(A_{1}\) and \(A_{2}, P\left(A_{1} \cup A_{2}ight)=\) \(P\left(A_{1}ight)+P\left(A_{2}ight)-P\left(A_{1} \cap A_{2}ight)\).

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Theorem 2.4 (Bonferroni). Let {A} be a sequence of events from a prob- ability space (, F, P). Then P ( A.) | A P(A). i=1 i=1