Question: There are three events: $A$ and $B$ and $C$. We know $P(A|B) = P(B|C) = 0.5$. Then
$P(A|C)$ should be?

Is it $0.5\cdot0.5=0.25$?

The question only provide the above information and the question provide the following choices.

$a. 0.25$

$b. 0.5$

$c. 1$

$d. 0$

$e.$ None of the above is correct

Answer: 
Why should you be able to tell? You only know how $A$ behaves in the presence of $B$, and how $B$ behaves in the presence of $C$. How could you possibly deduce how $A$ behaves in the presence of $C$, regardless of the presence or absence of $B$?

Examples to show that you can't:

If the events are all independent, then $\mathbb{P}(A \vert C) = \mathbb{P}(A) = \frac{1}{2}$.

If the event $C$ is independent of the others, then $\mathbb{P}(A \vert B) = \mathbb{P}(B) = \frac{1}{2}$, but $\mathbb{P}(A \vert C) = \mathbb{P}(A)$ need not be $\frac{1}{2}$: let $B$ be the event "I get heads on the first of two coins" and $A$ be the event "I get heads on both of the two coins".
You should know at least the probability of the single events to tell. If you'd know you could apply the *Bayes* theorem and solve.