Not sure if this helps, but you can try it:
Create a calculated field, let's say it is named "Count White"
COUNTD(IF [person_id] != 'White (Not Hispanic or Latino')
Then create another calculation which is:
You could also just write it all in one calculation.
If you cannot get this to work, please post a sample packaged workbook (.twbx) and someone can help you set it up. It is sometimes difficult to answer questions based on a text description of the problem only.
I am not sure you what kind of output you want. Why don't you drag person_id, ethinic_category to row and column. then use quick table calculation, to get percentage of total. if you only want white, then drag the ethinic category to the filter.