Read the original Bloom Filter paper from 1970. It covers all of this and it's very short. If the test for membership returns 'probably' then you do the more expensive test if you need to know for certain or you don't use one to begin with:

Space/Time Trade-offs in Hash Coding with Allowable Errors BURTON H. BLOOM

https://dl.acm.org/citation.cfm?doid=362686.362692