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For a probabilistic discrete choice model with independent reversed Gumbel distributed random costs, closed form expressions for the choice probabilities are known under the assumption that the variance is the same for all choice alternatives. This assumption is highly disputable in many cases in reality. In this paper, closed form expressions for the choice probabilities are derived for the case of independent Weibull distributed random costs. It is then necessary to assume that the variance of the Weibull distributed cost is a specific increasing function of its mean, which, however, may be a more natural assumption in many cases. The theoretical results are explained and exemplified in terms of the familiar stochastic user equilibrium (SUE) traffic assignment model.