A binary language belongs to $\mathrm{ZPP}$ if a polynomial-time probabilistic machine may answer $0$, $1$, or "don't know", every definite answer is correct, and a definite answer occurs with probability at least $2/3$ on every input. The time bound holds on every branch, including branches that answer "don't know".

This is the bounded-time, failure-allowed definition of zero-error polynomial time. Its familiar expected-polynomial-time characterization uses repeated independent trials until a definite answer is obtained; the equivalence of these characterizations is not a theorem of this submission. Here $none$ denotes "don't know" and $some b$ a definite answer.