This is a question about probability theory. One often sees a probability measure characterised as being additive "for disjoint $X,Y$ we have: $P(X \cup Y) = P(X) + P(Y)$ " but sometimes as being additive like so: $P(X \cup Y) + P(X \cap Y) = P(X) +P(Y)$ . I guess what I'm asking is, how do you prove the inclusion/exclusion version of additivity from the version for disjoint sets. (The converse is trivial).
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
Split $X$ into the disjoint union of $X \cap Y$ and some other set $A$, and split $Y$ into the disjoint union of $X \cap Y$ and some other set $B$.
$X = (X\setminus Y) \cup (X\cap Y)$, therefore $P(X) = P(X\setminus Y) + P(X\cap Y)$.