  We compute the expectation of the number of linear spaces on a random
complete intersection in $p$-adic projective space. Here "random" means that
the coefficients of the polynomials defining the complete intersections are
sampled uniformly form the $p$-adic integers. We show that as the prime $p$
tends to infinity the expected number of linear spaces on a random complete
intersection tends to $1$. In the case of the number of lines on a random cubic
in three-space and on the intersection of two random quadrics in four-space, we
give an explicit formula for this expectation.
