I think the main bit your explanation is missing is how a bloom filter uses multiple hash functions to reduce collisions. For example, a bloom filter might have

I think the main bit your explanation is missing is how a bloom filter uses multiple hash functions to reduce collisions. For example, a bloom filter might have 3 hashes, and all of them have to hit for a key to be known to be in the set. This reduces the likelihood of false positive collisions while keeping the no false negatives guarantee.