Most explanations of Bayes miss the mark. Consider the following for the role of Pr(B).
The crux of Bayes is the "update factor" $[Pr(B|A) / Pr(B)]$.
This is the transformation applied to the prior.
If B always occurs in all states of the world, there is no information content & the update factor is 1.
In this case, $Pr(A|B) = Pr(A)$.
However, if B occurs frequently when A has occurred, but the overall probability of B occurring is very low, then there is high information content with respect to Pr(A).
The update factor will be HIGH and so $Pr(A|B) >> Pr(A)$.
For completeness, if B occurs rarely when A has occurred, but the overall probability of B occurring is very high, then there is also information content with respect to Pr(A), but in the opposite direction.
The update factor will be LOW and so $Pr(B|A) << Pr(A)$.
Purely mechanical explanations of Bayes seem to miss the genius of this simple equation.