possible values of $P(A|C)$ when we know that $P(A|B)=P(B|C) = 1/2$ What are possible values of $P(A|C)$ when we know that $P(A|B)=P(B|C) = \frac12$ ?  

It is fairlt difficult for me, and I can't deal with it.  I know conditional probablity and bayesian probablity, however it doesn't help me.  Could you help me ?
Fix $t\in[0,1]$.  We can associate the sets $(A\cup B\cup C)^\complement$, $A\setminus (B\cup C)$, $B\setminus (C\cup A)$, $C\setminus (A\cup B)$, $(B\cap C)\setminus A$, $(C\cap A)\setminus B$, $(A\cap B)\setminus C$, and $A\cap B\cap C$ with the following probabilities: $$0\,,\,\, 0\,,\,\, \frac{t}{3}\,,\,\, \frac{1-t}{3}\,,\,\, \frac{1-t}{3}\,,\,\, \frac{t}{3}\,,\,\, \frac{1-t}{3}\,,\text{ and  }\frac{t}{3}\,,$$ respectively.  Then, $\text{Prob}(A|B)=\text{Prob}(B|C)=\frac{1}{2}$ and $\text{Prob}(A|C)=t$.
Hint:
use the Law of total probability,
https://en.wikipedia.org/wiki/Law_of_total_probability

and consider stochastic independence (if given).
Answer for more information^^