  A strategy to evaluate the distribution of the largest Schmidt eigenvalue for
entangled random pure states of bipartite systems is proposed. We point out
that the multiple integral defining the sought quantity for a bipartition of
sizes N, M is formally identical (upon simple algebraic manipulations) to the
one providing the probability density of Landauer conductance in open chaotic
cavities supporting N and M electronic channels in the two leads. Known results
about the latter can then be straightforwardly employed in the former problem
for both systems with broken ({\beta} = 2) and preserved ({\beta} = 1) time
reversal symmetry. The analytical results, yielding a continuous but not
everywhere analytic distribution, are in excellent agreement with numerical
simulations.
