Work out proof of work time instantly with clear inputs, formula shown and shareable results.
A proof of work requiring b leading zero bits succeeds with probability 2^-b per attempt, so the expected number of hashes is 2^b. Because attempts are independent, the time is exponentially distributed: reaching 95 percent confidence takes about three times the mean, which is why individual solve times vary so widely around the expectation.
Proof-of-work timing
P(success per hash) = 2^-b; expected hashes = 2^b; hashes for confidence c = 2^b x ln(1/(1-c))
The success probability halves with every additional required zero bit, so expected attempts double. Difficulty is exponential in the bit count.
Anti-abuse tokens such as Hashcash for email and client puzzles for rate limiting, where a few hundred milliseconds of client work is negligible for one request and prohibitive for millions.