Given three sets $A$, $B$, and $C$ that $A$ and $B$ are independent and $A$ and $C$ are dependent, is it possible for $B$ and $C$ to be dependent? IOW, is there counterexample for the following formula or can we prove it?
$$(A \perp B) \wedge (A \not\perp C) \Longrightarrow B \perp C$$
Yes: for example take $A$ and $B$ to be number of heads from a toss of a fair coin and $C=A+B$
I.e. $P(A=0)=P(A=1)=\frac12$ and independently $P(B=0)=P(B=1)=\frac12$
So $P(C=2 \mid A=1)=\frac12$ but $P(C=2\mid A=0)=0$ and $P(C=2)=\frac14$ so $A$ and $C$ are not independent
Similarly $P(C=2 \mid B=1)=\frac12$ but $P(C=2\mid B=0)=0$ and $P(C=2)=\frac14$ and $B$ and $C$ are also not independent