A is independent of B, and B depends on A? I learned in class that the following three conditions are equal:


*

*$P(A\cap B)=P(A)\cdot P(B)$

*$P(A|B)=P(A)$

*$P(B|A)=P(B)$
But suppose event $A$ is a subset of $B$ such that the (2) condition applies and $P(B)<1$. In that case the probability $P(B|A)$ must equal $1$, because $A$ is a subset of $B$, but $P(B)< 1$, in contradiction to the assumption that (2) and (3) are equal, what am I missing?

If $A$ is a subset of $B$ and $P(B)< 1$, then all conditions are wrong. For (2), you get $P(A|B) = P(A)/P(B)\not=P(A)$.
