We consider families of skew-product maps, representing systems evolving in discrete time in which, at each time step, one of a number of transformations is chosen according to an i.i.d process and applied. We extend the notion of matching for such dynamical systems and we show that, for a certain family of piecewise affine random maps of the interval, the property of random matching implies that any invariant density is piecewise constant. We give an application by introducing a one-parameter family of random maps generating signed binary expansions of numbers. This family has random matching for Lebesgue almost every parameter, producing matching intervals that are related to the ones obtained for the Nakada continued fraction transformations. We use this property to study the expansions with minimal weight. Joint with K. Dajani, and C. Kalle.