  Let $P(d)$ be the probability that a random 0/1-matrix of size $d \times d$
is singular, and let $E(d)$ be the expected number of 0/1-vectors in the linear
subspace spanned by d-1 random independent 0/1-vectors. (So $E(d)$ is the
expected number of cube vertices on a random affine hyperplane spanned by
vertices of the cube.)
  We prove that bounds on $P(d)$ are equivalent to bounds on $E(d)$: \[ P(d) =
(2^{-d} E(d) + \frac{d^2}{2^{d+1}}) (1 + o(1)). \] We also report about
computational experiments pertaining to these numbers.
