Given two events $A$ and $B$ such that $P(A)=P(B)=P(A \mid B^c)=\frac{1}{3}$, are $A$ and $B$ independent?
I'm having trouble figuring out how I can manipulate these equations to see if they're independent. Of course, I just need to check to see if $P(A \cap B)=\frac{1}{9}$ but most of the formulas I can think of involving the intersection of two events assumes independence. I was thinking about how $P(A)=P(A \mid B^c)$ implies that $A$ and $B^c$ are independent, but that isn't enough to make the leap.
$P(A\cap B)=P(A)-P(A\cap B^{c})=P(A)- P(A|B^{c}) (1-P(B))=\frac 1 3 -\frac 2 9=\frac 1 9=P(A)P(B)$