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This paper is concerned with the differences between deterministic and randomized finite-memory decision rules. It is shown that for any hypothesis-testing problem there exists a such that, for any , the optimal deterministic rule with bits in memory has a lower error probability than the optimal randomized rule with bits in memory. Suboptimal deterministic rules with this property are demonstrated. These deterministic rules lose at most bits. Thus for large memories the fraction of memory, measured in bits, lost by deterministic rules is negligible.