Transitively in Conditional Probability

Question

Let $A$, $B$ and $C$ are events with $P(A) >0$, $P(B) >0$ and $P(C) >0$. Then show that

$P(A\mid B) < P(A)$ and $P(B\mid C) < P(B)$ do not imply that $ P(A\mid C) < P(A)$

I have tried several counterexamples to disprove the above but it could not work out with me. Not sure what to do with it. Any hint I would very much appreciate it.

Answer 1

Just take any two disjoint events $A,B$ with $P(A)>0$ and $P(B)>0$ take $C=A$. Note that $P(A|C)=1$ so $P(A|C)<P(A)$ does not hold. I will let you write down an explicit example.