# Determining independence of events

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.

$$P(A\cap B)=P(A)-P(A\cap B^{c})=P(A)- P(A|B^{c}) (1-P(B))=\frac 1 3 -\frac 2 9=\frac 1 9=P(A)P(B)$$