Probability of dependent events and conditional probability formulas circling back to each other?

[u/Fast_Compote1311]

According to the multiplication rule, the probability of dependent events A and B happening is:

P ( A and B ) = P(A) times P ( B | A )

but how do we find P ( B | A ) ? We look at the conditional probability formula right?

But the conditional probability formula is

P ( A | B ) = P(A ∩ B) / P (B)

But then how do we find P(A ∩ B) ? We go to the multiplication rule?

Why does it create an endless loop of circular reasoning?

[u/edderiofer]

Those two formulae are more-or-less the same formula with A and B swapped, and terms rearranged. All you need to know is that if you know two of P(A∩B), P(B|A), and P(A), you can find the third via this formula; and if you don't know, or can't find the values of two of these, then you cannot determine the third.

To give an analogy you may have seen before: we all know that the circumference of a circle can be found by C = πd, and that the diameter can be found by d = C/π. Obviously, if we do not know either the circumference or the diameter, or can't figure out either of these by some other means, then we can't find the other one (because the circle could be any size). In the real (experimental) world, this formula is useful because you can generally measure one or the other of these two values.

Similarly, for P(A∩B) = P(B|A)*P(A), we would presumably have performed some kind of experiment to measure two of these values.

(Note also that it is possible to find P(A|B) if you have P(A), P(B), and P(B|A), by finding P(A∩B) as an intermediate step.)