Set Operations in Probability

Events are sets, so the algebra of sets is the algebra of events. The operations are simple; their value is that they let you rewrite an event you cannot compute into one you can.

The three operations

The union $A \cup B$ contains everything in $A$ , in $B$ , or in both. Read it as " $A$ or $B$ ", with the inclusive "or".

The intersection $A \cap B$ contains only what is in both. Read it as " $A$ and $B$ ".

The complement $A^{\mathsf{c}}$ contains everything in $\Omega$ that is not in $A$ . Read it as "not $A$ ".

Two sets are disjoint when $A \cap B = \emptyset$ , which is the set-theoretic name for mutually exclusive. This is the condition that lets probabilities add.

Translating between English and these symbols is a real skill under pressure. "At least one", "neither", "exactly one" and "at most one" all describe different sets, and mixing them up is a common way to answer a well-understood problem incorrectly.

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