In the correct definition of randomized space-bounded computation, the machine has access to a random coin. The coin can be flipped at will, but outcomes of previous coin flips cannot be recalled unless they are saved in the machine's limited memory. In contrast to this read-once mechanism of accessing the random source, one may consider Turing machines which have access to a random tape. Here, the random bits may be multiply accessed by the machine. The author demonstrates a very concrete sense in which multiple access to the random bits is better than read-once access to them: Every language accepted with bounded two-sided error by a read-once-randomized logspace machine can be accepted with zero error by a randomized logspace machine having multiple access to the random bits.