Regarding superfluous division by P(A): it is omitted in some applications of Bayesian reasoning.

Search theory begins with a probability map and updates the map based on searching and not finding. The calculations are done unnormalized both for speed and to preserve information. If the map started with 10,000 points distributed over all the regions in the search area, it is useful to know that searching has reduced the total to 1,000. It suggests you have exhausted the area and should reconsider your hypothesis that the subject/object/target is in this area.

There are equivalent approaches using normalization, such as including "Rest of World" as an area. But unnormalized is faster.