Apologies in advance, I'm new to spatial analyst...

I have several irregularly-shaped zones, each with a count indicating the number of trips made in that particular zone. I would like these trip totals to be distributed in each zone based on the distribution of points exhibited in a separate point layer.
Could someone point me in the right direction in terms of the tools to use? It seems like the "zonal" tools aren't really what I need.

Seems like 2 main steps:

1) Create raster based on the point layer (counting the number of points in each cell). This is my distribution layer.

arcpy.sa.PointDensity(...) ?

2) Create a new raster whereby the cells within each zone add up to the total "count" for that zone.

arcpy.sa.ExtractByPolygon(...)
&
arcpy.sa.Times(...)
&
...

???


Thanks !


btw, I don't need to get fancy with accounting for the polygon boundary cells - results don't need to be super accurate.