Assume that NP is not included in BPP. Gutfreund, Shaltiel, and Ta-Shma in [Computational Complexity 16(4):412-441 (2007)] have proved that for every randomized polynomial time decision algorithm D for SAT there is a polynomial time samplable distribution such that D errs with probability at least 1/6-epsilon
on a random formula chosen with respect to that distribution. In this paper, we show how to increase the error probability to 1/3-epsilon.

Removed the result on generating hard instances for NP search problems, as it has been published in a paper of Gutfreund (2006) of which I was unaware.

Assume that $NP\ne RP$. Gutfreund, Shaltiel, and Ta-Shma in [Computational Complexity 16(4):412-441 (2007)] have proved that for every randomized polynomial time decision algorithm $D$ for SAT there is a polynomial time samplable distribution such that $D$ errs with probability at least $1/6-\epsilon$ on a random formula chosen with respect to that distribution. In this paper, we show how to increase the error probability to $1/3-\epsilon$. We also generalize this result to the search version of SAT: we prove that for every randomized polynomial time algorithm $S$ searching for a satisfying assignment of a given formula, there is a polynomial time samplable distribution such that $S$ errs with probability at least $1-\epsilon$ on a random formula chosen with respect to that distribution.