  Let $\mu$ be a Gaussian measure (say, on ${\bf R}^n$) and let $K, L \subset
{\bf R}^n$ be such that K is convex, $L$ is a "layer" (i.e. $L = \{x : a \leq <
x,u > \leq b \}$ for some $a$, $b \in {\bf R}$ and $u \in {\bf R}^n$) and the
centers of mass (with respect to $\mu$) of $K$ and $L$ coincide. Then $\mu(K
\cap L) \geq \mu(K) \cdot \mu(L)$. This is motivated by the well-known
"positive correlation conjecture" for symmetric sets and a related inequality
of Sidak concerning confidence regions for means of multivariate normal
distributions. The proof uses an apparently hitherto unknown estimate for the
(standard) Gaussian cumulative distribution function: $\Phi (x) > 1 -
\frac{(8/\pi)^{{1/2}}}{3x + (x^2 +8)^{{1/2}}} e^{-x^2/2}$ (valid for $x > -1$).
