  It is shown that a large class of events in a product probability space are
highly sensitive to noise, in the sense that with high probability, the
configuration with an arbitrary small percent of random errors gives almost no
prediction whether the event occurs. On the other hand, weighted majority
functions are shown to be noise-stable. Several necessary and sufficient
conditions for noise sensitivity and stability are given.
  Consider, for example, bond percolation on an $n+1$ by $n$ grid. A
configuration is a function that assigns to every edge the value 0 or 1. Let
$\omega$ be a random configuration, selected according to the uniform measure.
A crossing is a path that joins the left and right sides of the rectangle, and
consists entirely of edges $e$ with $\omega(e)=1$. By duality, the probability
for having a crossing is 1/2. Fix an $\epsilon\in(0,1)$. For each edge $e$, let
$\omega'(e)=\omega(e)$ with probability $1-\epsilon$, and
$\omega'(e)=1-\omega(e)$ with probability $\epsilon$, independently of the
other edges. Let $p(\tau)$ be the probability for having a crossing in
$\omega$, conditioned on $\omega'=\tau$. Then for all $n$ sufficiently large,
$P\{\tau : |p(\tau)-1/2|>\epsilon\}<\epsilon$.
