Several methods of watermark detection involve computing a vector from some input media, computing the normalized correlation between that vector and a predefined watermark vector, and comparing the result against a threshold. We show that, if the probability density function of vectors that arise from random, unwatermarked media is a zero-mean, spherical Gaussian, then the probability that such a detector will give a false detection is given exactly by a simple ratio of two definite integrals. This expression depends only on the detection threshold and the dimensionality of the watermark vector.