Does it matters if we swap the position of random variables appearing in the expression? Recall P(A AND B) = P(A|B)P(B) = P(B|A)P(A). Mathematically, it seems as long as all the terms in the expression is given, the position of A and B doesn't matter much. (i.e. the line defining which variable acts as 'condition' is blurred). However, logically sometimes it doesn't make sense. E.g. when the ground of the street is wet, it is because of raining. Here, P(wet|rain) defined but P(rain|wet) not, so I highly doubt how can 'P(A AND B) = P(A|B)P(B) = P(B|A)P(A)' stand in this case? Or it would stand all the time as long as the number of each term is given, without consideration to the logic of the true application? Or it's just me getting the whole thing wrongly? Please advise and thanks.


Here, P(wet|rain) defined but P(rain|wet) not, so I highly doubt how can 'P(A AND B) = P(A|B)P(B) = P(B|A)P(A)' stand in this case?

Probability does not imply a causality relation. P(wet|rain) gives a probability that something is wet, given that it rained. P(rain|wet) is the probability that it rained, provided that something is wet - e.g., you go out into the street and see that the pavement is wet, which could be because it rained during the night or because the street was washed by a cleaning truck.
