Let $A$ be event and probability $\mathbb{P}(A)$ is $0$ or $1$. How to show that two events $A$ and $B$ are independent of each other. Here $B$ is any other event.
So I think I need to proove $\mathbb{P}(B|A)= \mathbb{P}(A)$ or what?. How to start and what to do?
If $P(A)=0$ then $P(A)P(B)=0$ and $P(A\cap B)\leq P(A)=0$ so $P(A\cap B)=P(A)P(B)$. If $P(A)=1$ then $P(A)P(B)=P(B)$ and $P(A\cap B)=P(B)$ becasue $P(B)=P(A \cap B)+P(B\setminus A)$ and the second term is $0$. [ $P(B\setminus A)\leq P(A^{c})=1-P(A)=0$].