How do I understand "time" in conditional probability and chain rule

Question

P(ABCD)=P(A|BCD)P(B|CD)P(C|D)P(D)

I really do not get this to be true, the reason is that the conditional statement say: Given this has happened already (If I am correct). So it tells something about the TIME when certain events occur in relation to other events.

Example: P(A|B) = probability for A to occur given B already has occurred or (B occure BEFORE A).

P(ABCD) = The same as the intersection of all the events happening, but tells nothing about certain events happening before others.

Any comments about this that may clarify, thank you for reading?

Answer 1

So for the equation $$P(ABCD) = P(A|BCD)P(B|CD)P(C|D)P(D)$$ the RHS, reading from right-to-left, is

The probability that $D$ occurred,

times the probability that $C$ occurred, assuming knowledge (i.e., it's given that) $D\;$occurred,

times the probability that $B$ occurred, assuming knowledge that $C,D\;$both occurred,

times the probability that $A$ occurred, assuming knowledge that $B,C,D\;$all occurred.

But there is only one trial, no time sequence.

Each conditional probability represents a probability which takes into account the information provided (i.e., assuming the known occurrence of the events to the right of the vertical bar).

Answer 2

Yes, you are right. For two events $A$ and $B$: $$P(A\cap B)=P(A)\cdot P(B|A) \ \ (A, \ then \ B)$$ $$P(B\cap A)=P(B)\cdot P(A|B) \ \ (B, \ then \ A)$$ However, in the given formula, the information about the sequence of events is also indicated. Start interpreting from the end: $$P(C|D) \ (D, \ then \ C)$$ $$P(B|C\cap D) \ (C\cap D, \ then \ B)=(D, \ then \ C, \ then \ B)$$ $$P(A|B\cap C\cap D) \ (B\cap C\cap D, \ then \ A)=\cdots=(D, \ then \ C, \ then \ B, \ then \ A).$$