Is conditional probability transitive? Is conditional probability "transitive" (or how else this is called - please, explicate) in the meaning that $P(a \mid c)=P(a \mid b)P(b \mid c)$ ? Intuitively this seems so, but could you comment/proove on how to understand this (or refute)?

From the definition of conditional probability, we have
$$
P(A\mid B)P(B\mid C) = \frac{P(A\cap B)}{P(B)} \frac{P(B\cap C)}{P(C)}
$$
and
$$
P(A\mid C) = \frac{P(A\cap C)}{P(C)}.
$$
It's not clear to me why these should be equal. Let's find a counterexample.
Suppose we roll a fair die. Let $A$ be the event that $1$ shows, let $B$ be the event that an even number appears, and let $C$ be the event that either $1$, $2$, or $3$ shows.
Then
$$
P(A\mid C) = 1/3
\quad\text{and}\quad
P(A\mid B) = 0
\quad\text{and}\quad
P(B\mid C) = 1/3.
$$
This example shows that the desired formula does not hold in general.
