On read once vs. multiple access to randomness in logspace

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. In this note we show 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 2-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. Finally, we characterize the clsss of languates that can be accepted with two-sided error by randomized logspace machines with multiple access to the random bits as exactly the class of languages that are in logspace to almost every oracle.