Corollaries

Adding non-disjoint events:

$$\mathbb{P}[A \cup B] = \mathbb{P}[A] +\mathbb{P}[B] - \mathbb{P}[A\cap B]$$

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Union bound: $$\mathbb{P}[A \cup B] \leq \mathbb{P}[A] +\mathbb{P}[B]\\ $$ The union bound is frequently used for analyzing problems where the intersection is difficult to evaluate (e.g. multiple hypothesis testing)