Singular 0/1-matrices, and the hyperplanes spanned by random 0/1-vectors

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 $E(d)$: $$P(d)=(2^{-d}E(d)+\frac{d^2}{2^{d+1}})(1+o(1))\mathrm{.}$$ We also report about computational experiments pertaining to these numbers.