A number of probabilistic, physical models of the process of fatigue damage have been proposed in the literature. Each leads to a different distribution of the number of load applications to failure. These include the Wei bull distribution, the lognormal distribution, and the gamma distribution. Other distributions exist (see, for example, Rascon [1967]), but several can be excluded on physical grounds, and others can be excluded because of their lack of ability to "fit" statistical observations. Remaining contending models have better or poorer physical arguments, and better or poorer "fits" to data. An expert in the field has summarized the situation at a particular point in time by saying, "Neither the Wei bull nor gamma distributions can be excluded. If I had to give them relative weights now, I would say perhaps 60 to 40 Wei bull over gamma." Two designs for a structural member are under consideration. If alternative a is analyzed assuming that the Wei bull distribution holds, it has a reliability with respect to fatigue distress during its lifetime of 0.98, but the probability is 0.99 under the gamma model. On the other hand, the parameters are such that for the second design the reverse is true. The probability of failure is 0.01 under the Wei bull hypothesis and 0.02 under the gamma hypothesis. (It is not known that these precise numbers could be constructed with these two particular distributions, but the situation is conceptually important even if there is only a change in the relative values of the probabilities.) If the designs are of comparable initial cost and if the cost of failure is the same in both cases, which design should the designer use (to reflect best the information available to him)?