Explain Bloom filters, their false positive rate, and applications.

Bloom filters are space-efficient probabilistic structures testing set membership. They use a bit array with multiple hash functions. Insert: set bits at hash positions. Query: check all hash positions - if all set, probably present (possible false positive); if any unset, definitely absent (no false negatives). False positive rate is approximately (1-e^(-kn/m))^k where k=hash functions, n=elements, m=bits. Used in caching, network routers, and databases to avoid expensive lookups.