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A randomized algorithm makes some choices using random bits.

For a fixed input, the random bits can change the path taken by a randomized algorithm, its output, or its running time. Specify the distribution of those choices and which guarantee is required: one-sided error permits mistakes of only one kind, whereas ZPP requires a correct answer and bounds the expected running time. Polynomial identity testing makes this distinction concrete: choosing different field elements changes whether a nonidentity is detected.

A uniform distribution on a finite set must be implemented faithfully. To choose uniformly from $\{0,1,2,3,4\}$ using fair independent bits, read three bits as a number from zero to seven, discard five through seven, and repeat until a permitted value appears. Each retained value has the same chance. Taking the three-bit number modulo five instead would give zero, one and two twice as many preimages as three and four, invalidating a calculation that assumes equal chances.