I've encountered a question which includes a chain of events and their probabilities. The picture describes the scenario:
I want to know the Probability of $P(P_3|P_1)$.
my approach was as follows:
$P(P_3|P_1)$= $P(P_3|P_2,P_1)\times P(P_2) + P(P_3|\bar{P_2},P_1)\times P(\bar{P_2})$.
$P(P_3|P_2,P_1) = \frac{P(P_1,P_2,P_3)}{P(P_2,P_1)}$
and for $P(P_3|\bar{P_2},P_1) = \frac{P(P_1,\bar{P_2},P_3)}{P(\bar{P_2},P_1)}$.
$P(P_2) = P(P_2|P_1)\times P(P_1) + P(P_2|\bar{P_1})\times P(\bar{P_1})$.
$P(P_2,P_1) = P(P_2|P_1)\times P(P_1)$ and $P(\bar{P_2},P_1) = P(\bar{P_2}|P_1)\times P(P_1)$ and $ P(\bar{P_2}|P_1) = 1 - P(P_2|P_1) $
Can anyone guide me If I went wrong in my reasoning ? Shall I include $P_4$ event in the derivation? I didn't include it since $P_3$ and $P_4$ are conditionally independent events.
Thanks