arcgis desktop - Assign raster cells presence/absence values depending on whether polygon overlaps with cells

I have a raster of continuous numeric values. A certain portion of said raster overlaps with a polygon of interest. I want to know which cells overlap, and assign values to the raster's attribute table corresponding to whether or not there is overlap (e.g. 0 for no overlap with polygon, 1 for overlap).

Ideally it would be great to be able to set an overlap threshold so that only raster cells where there is more than 50% overlap are assigned a "1". Not a must, but would be very helpful.