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We extend the notion of matching for one-dimensional dynamical systems torandom matching for random dynamical systems on an interval. We prove that fora large family of piecewise affine random systems of the interval the propertyof random matching implies that any invariant density is piecewise constant. Wefurther introduce a one-parameter family of random dynamical systems thatproduce signed binary expansions of numbers in the interval [-1,1]. This familyhas random matching for Lebesgue almost every parameter. We use this to provethat the frequency of the digit 0 in the associated signed binary expansionsnever exceeds 1/2.

Matching for random systems with an application to minimal weight expansions