  We study random, finite-dimensional, ungraded chain complexes over a finite
field and show that for a uniformly distributed differential a complex has the
smallest possible homology with the highest probability: either zero or
one-dimensional homology depending on the parity of the dimension of the
complex. We prove that as the order of the field goes to infinity the
probability distribution concentrates in the smallest possible dimension of the
homology. On the other hand, the limit probability distribution, as the
dimension of the complex goes to infinity, is a super-exponentially decreasing,
but strictly positive, function of the dimension of the homology.
