I always found memorizing the formula for Bayes theorem and the various symbols very tough. So here's how I used to solve the problems. Using just basic Venn diagrams, or set diagrams, you can solve virtually any problem of conditional/combined probability.

First an introduction to Venn diagrams.

In a Venn diagram, you denote the sample space or the universal set by a rectangle. This signifies all the available options to the value. Of course this is just symbolic.

In the sample space, usually you denote the different events by circles.

Now if you're given two events A and B and are asked to find the conditional probability P( A | B ), which means the probability A happens given that B happens, here's how you proceed. Here's the Venn diagram for it.

Here the yellow circle denotes event A, and the blue circle event B. The area common to both, is denoted by green. This is AB where denotes intersection. Now to find the probability that A happens given that B happens.

When you're given the condition that B happens, your sample space becomes B. That is, instead of considering the whole rectangle as the sample space, you only consider the blue circle.

Now the area for event A the given sample space is AB.

So the probability for event A is P (AB) / P(B)

Therefore, P ( A | B ) = P (AB) / P(B)

Similarly, for P ( B | A ) = P (AB) / P(A)

Eliminating P (AB),

P (A | B) = P( B | A ).P(A) / P(B)

A similar approach can be used for other probability questions.

Source is me. Had to figure all this out myself when I was learning, didn't find it in any textbook. So I thought it might be useful to some people. Sorry if this has been a waste of time :D. Thanks for reading through