Checking the independence of events

Question

Let events A and B be independent such that 0 < P(A) < 1 and 0 < P(B) < 1.

How should I check the independence of events C = A - B and D = B - A?

Answer 1

Recall that events $E$ and $F$ are independent precisely if $\Pr(E\cap F)=\Pr(E)\Pr(F)$.

So we want to check whether $\Pr(C\cap D)=\Pr(C)\Pr(D)$.

We have $C\cap D=(A\setminus B)\cap (B\setminus A)$. But $(A\setminus B)\cap (B\setminus A)=\emptyset$ (the empty set), so has probability $0$.

We have shown that $\Pr(C\cap D)=0$. If we can show that $\Pr(C)$ and $\Pr(D)$ are both non-zero, we will have shown that $\Pr(C\cap D)\ne \Pr(C)\Pr(D)$, meaning that $C$ and $D$ are not independent.

We have $\Pr(A\setminus B)=\Pr(A)-\Pr(A\cap B)=\Pr(A)-\Pr(A)\Pr(B)=\Pr(A)(1-\Pr(B))$. Since $\Pr(A)\ne 0$ and $\Pr(B)\ne 1$, we conclude that $\Pr(A\setminus B)\ne 0$.

A similar calculation shows that $\Pr(B\setminus A)\ne 0$.