  We study the expected behavior of the Betti numbers of arrangements of the
zeros of random (distributed according to the Kostlan distribution) polynomials
in $\mathbb{R}\mathrm{P}^n$. Using a random spectral sequence, we prove an
asymptotically exact estimate on the expected number of connected components in
the complement of $s$ such hypersurfaces in $\mathbb{R}\mathrm{P}^n$. We also
investigate the same problem in the case where the hypersurfaces are defined by
random quadratic polynomials. In this case, we establish a connection between
the Betti numbers of such arrangements with the expected behavior of a certain
model of a randomly defined geometric graph. While our general result implies
that the average zeroth Betti number of the union of random hypersurface
arrangements is bounded from above by a function that grows linearly in the
number of polynomials in the arrangement, using the connection with random
graphs, we show an upper bound on the expected zeroth Betti number of random
quadrics arrangements that is sublinear in the number of polynomials in the
arrangement. This bound is a consequence of a general result on the expected
number of connected components in our random graph model which could be of
independent interest.
