Consider an Ellsberg experiment in which one can win by calling the color (red or blue) of the ball that will be drawn from an urn in which the balls are of unknown proportions. It is well known (yet rarely advertised) that selecting the color based on a fair sided coin completely eradicates the ambiguity about the odds of winning. In this talk, we explore what are conditions under which a decision maker that employs a risk measure should have his action depend on the outcome of an independent random device. Surprisingly, we show that for any ambiguity averse risk measure there always exists a decision problem in which a randomized decision strictly dominates all deterministic decisions.