TITLE: If $P(A \ \cup \ B) = P(A) + P(B)$, is it the case that $A$ and $B$ are disjoint?
QUESTION [7 upvotes]: I know that if $A$ and $B$ are disjoint events, then $P(A \cup \ B) = P(A) + P(B)$. However, is the converse true as well? Thanks.

REPLY [7 votes]: No for example suppose you have a uniform distribution on $[0,1]$.  Let $A=[0,1/2]$ and $B=[1/2,1]$.  Then $P(A\cap B)=0$ but $A\cap B=\{1/2\}\not=\emptyset$.