For every e>0, we present a log-space deterministic algorithm
that correctly decides undirected graph connectivity
on all but at most exp(n^e) of all n-vertex graphs.
The same holds for every problem in Symmetric Log-space (SL).

Using a plausible complexity assumption (which is not known to imply BPP=P)
we show that, for every e>0, each problem in BPP has a polynomial-time
deterministic algorithm that errs on at most exp(n^e) of all n-bit long inputs.
The same holds if we replace BPP by MA and deterministic by
nondeterministic algorithms.

The results are obtained by exploring which probabilistic
algorithms can be derandomized (in the sense above) by generating
their coin tosses deterministically from  the input itself.
--------------------------------------------------------------------------