Proof of Markov Property

Question

I'm trying to understand a simple proof for the markov property which states that:

"$A_1$, $A_3$ are conditionally independent given $A_2$ iff $P(A_3 | A_1 \cap A_2)=P(A_3|A_2)$"

The Proof begins as follows:

"only if"

$$P(A_3 | A_1 \cap A_2) = \dfrac{P(A_1 \cap A_2 \cap A_3)}{P(A_1 \cap A_2)} \\ = \dfrac{P(A_1 \cap A_3 | A_2)P(A_2)}{P(A_1 \cap A_2)} \\ = \dfrac{P(A_3|A_2)P(A_1 A_2)P(A_2)}{P(A_1 \cap A_2)}$$

then the proof ends. I'm not really sure why this proves it?

If a question was asked "Prove that for events $A_1$; $A_2$; $A_3$ $\in F$, such that $A_1$ and $A_3$ are conditionally independent given $A_2$ iff the Markov property holds. Would this suffice as a proof?

Answer 1

To finish off the proof: $$\dfrac{P(A_3|A_2)P(A_1|A_2)P(A_2)}{P(A_1 \cap A_2)}=\dfrac{P(A_3|A_2)P(A_1 \cap A_2)}{P(A_1 \cap A_2)}=P(A_3|A_2)$$

Answer 2

If $P(A_1 A_2)$ is actually $P(A_1 |A_2)$ then you have $$ \frac{P(A_3|A_2)P(A_1 \cap A_2)}{P(A_1 \cap A_2)} = P(A_3 | A_2) $$