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Independent repetitions reduce a randomized algorithm's error probability while preserving polynomial running time.

The combining rule depends on the error orientation. For one-sided error with false acceptance probability at most $p<1$, reject as soon as any trial rejects and accept only if every trial accepts. For $k$ independent trials on the same fixed negative input, false acceptance has probability at most $p^k$. The protected positive inputs still always accept. If false rejection is the possible mistake instead, accept as soon as any trial accepts. This uses the decisive answer rather than a majority vote.

For constant $p<1$, $k$ grows only logarithmically in $1/\epsilon$ to achieve error at most $\epsilon$. A polynomial number of trials preserves a polynomial-time algorithm's running-time bound. Independence matters: repeating one answer, or evaluating at the same random point each time, supplies no new evidence. The independent repetition of a polynomial identity test gives an exact $25$-outcome example.