The Biased Coin Problem

A slightly random source (with bias$\epsilon $) is a sequence $\mathbf{x} = (\mathbf{x}_1 ,\mathbf{x}_2 , \ldots ,\mathbf{x}_n )$ of random bits such that the conditional probability that $\mathbf{x}_i = 1$, given the outcomes of the first $i - 1$ bits, is always between $\frac{1}{2} - \epsilon $ and $\frac{1}{2} + \epsilon $. Given a subset S of $\{ 0,1\} ^n $, define its $\epsilon $-biased probability to be the minimum of $\text{Pr}[ \mathbf{x} \in S ]$ over all slightly random sources $\mathbf{x}$ with bias $\epsilon $. It is shown that, for every fixed $\epsilon < \frac{1}{2}$ and almost every subset S of $\{ 0,1\} ^n $, the $\epsilon $-biased probability of S is bounded away from 0.