# Variations in Bayes' theorem in the denominator

I've got a question regarding the different variations of the denominator in Bayes' Theorem. I'm led to believe the denominator of this equation is = P(B)

I did some research and read about a rule pertaining to the 3rd axiom stating P(A) = summation of P(Ai,Bj) of mutually exclusive events. I also know P(A,B) = to P(B|A) * P(A). We can all agree though that if both events are mutually exclusive P(A,B) = 0. So wouldn't that just sum a bunch of 0's? Can someone please help explain this?

The way the sum rule of probability is generally stated is, $$P(A) = \sum_B P(A,B)$$
where the sum is over all possible mutually exclusive values of the variable $B$. Not mutually exclusive with $A$, but mutually exclusive with each other. So, if $b_k$ and $b_l$ are two possible values of $B$ then they are mutually exclusive with each other. That is, you cannot have $b_k$ and $b_l$ both in one observation if $k \ne l.$ But there's no assumption on either of their likelihood of appearing with any observation of $A$.
As an illustrative example, suppose you throw a dart at a dart board. The position on which the dart lands can be described by the radius, $r$, from the center, and the angle $\phi \in [0, 2 \pi)$. The probability distribution of its landing spot can be written as $P(r, \phi).$ Clearly you can't land on two different values of $\phi$, nor can you land on different values of $r.$ But there's no apparent restriction on a value of $r$, given a value of $\phi$, unless explicitly stated in the joint distribution. To get the probability distribution of the radius, we use the sum rule, summing the joint distribution over all values of $\phi$, $$P(r) = \int_0^{2 \pi} P(r, \phi) d \phi.$$