L01 · Sample spaces & events — Introduction to Probability

Probability axioms

A probability $P$ on a sample space $\Omega$ satisfies non-negativity, $P(\Omega)=1$ , and countable additivity for disjoint events.

If $A,B$ are events, prove $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ .

Solution · P1

Write $A\cup B$ as a disjoint union of $A\setminus B$ , $B\setminus A$ , and $A\cap B$ .
Apply finite additivity and rewrite using $P(A)=P(A\setminus B)+P(A\cap B)$ (and same for $B$ ).