Noise Stability of Weighted Majority
  Benjamini, Kalai and Schramm (2001) showed that weighted majority functions of $n$ independent unbiased bits are uniformly stable under noise: when each bit is flipped with probability $\epsilon$, the probability $p_\epsilon$ that the weighted majority changes is at most $C\epsilon^{1/4}$. They asked what is the best possible exponent that could replace 1/4. We prove that the answer is 1/2. The upper bound obtained for $p_\epsilon$ is within a factor of $\sqrt{\pi/2}+o(1)$ from the known lower bound when $\epsilon \to 0$ and $n\epsilon\to \infty$. 
Yuval Peres