  We express the Masur-Veech volume and the area Siegel-Veech constant of the
moduli space of meromorphic quadratic differential with simple poles as
polynomials in the intersection numbers of psi-classes supported on the
boundary cycles of the Deligne-Mumford compactification of the moduli space of
curves. Our formulae are derived from lattice point count involving the
Kontsevich volume polynomials that also appear in Mirzakhani's recursion for
the Weil-Petersson volumes of the moduli space of bordered hyperbolic Riemann
surfaces.
  A similar formula for the Masur-Veech volume (though without explicit
evaluation) was obtained earlier by Mirzakhani through completely different
approach. We prove further result: up to an explicit normalization factor
depending only on the genus and on the number of cusps, the density of the
orbit of any simple closed multicurve computed by Mirzakhani coincides with the
density of square-tiled surfaces having horizontal cylinder decomposition
associated to the simple closed multicurve.
  We study the resulting densities in more detail in the special case when
there are no cusps. In particular, we compute explicitly the asymptotic
frequencies of separating and non-separating simple closed geodesics on a
closed hyperbolic surface of genus g for all small genera g and we show that in
large genera the separating closed geodesics are exponentially less frequent.
  We conclude with detailed conjectural description of combinatorial geometry
of a random simple closed multicurve on a surface of large genus and of a
random square-tiled surface of large genus. This description is conditional to
the conjectural asymptotic formula for the Masur-Veech volume in large genera
and to the conjectural uniform asymptotic formula for certain sums of
intersection numbers of psi-classes in large genera.
