a) Show that for any events $A, B$, and $C$ we have $P(A\mid B) = P(A\mid B \cap C)P(C\mid B) + P(A\mid B \cap C')P(C'\mid B)$.
b) Prove or give a counterexample. If $A$ and $B$ are independent, then they are conditionally independent given $C$.
I am not strong with proofs by any means. Like I said, I know that P(C|B)=[P(C∩B)]/P(B) so maybe if I were to multiply through by P(B) then use P(C|B)=[P(C∩B)]/P(B)?
Consider a partition of the Sample Space $\Omega$ into $C,C^c$. Consider an Event $A$. The sets $A\cap C $ and $A\cap C^c$ are disjoint i.e. $(A\cap C)\cap(A\cap C^c)=\phi$. Hence by the 3rd Axiom of Probability Measures, $P(A) = P(A\cap C)+P(A\cap C^c)$. Likewise you can also prove the required question.
The example that I explained for Part b is reasonably simple. If $A,B$ take on values $\left\{0,1\right\}$ then $C$ also takes on the same values when $+$ is a Binary Ex-Or operation. If I tell you that $C$ has taken the value $1$ and also tell you that $A=1$, then $B$ cannot be anything other than $0$ and thus $A,B$ become dependent when the value of $C$ is known.