The problem of branch (or state) selection, initiated by a slowly changing external parameter in a bifurcating system has wide applications to far-from-equilibrium processes in Physics, Chemistry and Biology. In macroscopic systems such as laser and fluid dynamic transitions, bifurcating chemical reactions and possibly some selection processes in certain biological applications, for example, state selection always takes place in the presence of noise and a nonzero (though possibly quite small) symmetry breaking field. We consider a quite general representation of a transition from a monostable to a bistable state, driven by a linearly time varying control parameter in the presence of additive, Gaussian, white noise with a nonzero (though small compared to the noise variance) mean. We have constructed an electronic analog of this system, and developed methods for measuring the time evolution of its statistical densities (the first such measurements on a physical system). The probabilities of branch selection are extracted from these measurements and are compared to the predictions of a recent theory with applications to the problem of prebiotic selection of molecular chirality.