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?

It is not possible to have sets $A$ and $B$ with $A \subseteq B$, $P(A), P(B) > 0$ and $P(B) < 1$ that fulfill condition (ii).
Assume that such sets exist. Multiplying (ii) with $P(B)$ yields (i). Then we have
$$
P(A) = P(A \cap B) = P(A)\cdot P(B) < P(A),
$$
a contradiction.
