  We formulate and prove a Jakobson-Benedicks-Carleson type theorem on the
occurence of nonuniform hyperbolicity (stochastic dynamics) in families of
one-dimensional maps, based on "computable starting conditions" and providing
"explicit, computable," lower bounds for the measure of the set of selected
parameters. As a first application of our results we show that the set of
parameters corresponding to maps in the quadratic family f_{a}(x) = x^{2}-a
which have an absolutely continuous invariant probability measure is at least
10^-5000 !
