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A non-deterministic approach to engineering problem solving is developed, with the emphasis on the effects of manufacturing and service uncertainties on selected performance characteristics and economic factors. The approach integrates concepts from probability analysis, statistical inference, and decision making theory into a coherent framework, which is identified here as the Decision Model for Engineering (DME). The main innovative aspect of this approach is the application of a general method for statistical inference based on the Bayes' theorem to quantify probabilities of random variables for subsequent use as inputs to a decision making model. The solution to a Bayesian inference problem is obtained in terms of "posterior probabilities" of selected random variables, which are calculated by combining "prior probabilities" of the same variables with available test data. The "prior probabilities" are quantified in a "personal", or "subjective", manner by leveraging prior knowledge, experience, and intuition to estimate feasible or likely variations of selected parameters. This is in contrast to "classical" statistics, which does not support the concept of "prior" probability, since it quantifies the chances of occurrence of random events in terms of "relative frequency" probabilities, which are determined from large samples of actual test data. Unlike existing applications of the Bayes' theorem, the DME methodology does not treat the posterior probabilities as "final" results, but it uses them as inputs to a decision making model that identifies the "best" possible action, by singling out the alternative which minimizes a previously defined measure of "risk", or maximizes a previously defined measure of "utility". The example of analyzing the effects of possible fiber misalignment angles caused by variability in the manufacturing a batch of unidirectional composite laminae is used to demonstrate the applicability and the benefits of the DME concepts to solving practical engineering problems. The elastic constants and strengths properties of such laminae are predicted for various reasonable values of fiber misalignment angles. The Taguchi's quadratic cost function is, subsequently, used to assess the cost of the deviation of a property from the ideal case (0{dollar}\\sp\\circ{dollar} fiber angle). Additional methods of "Applied Mathematics", such as regression analysis, Monte Carlo simulation, and sensitivity analysis are utilized, respectively, to predict unknown variables, generate required data, and study the variations of laminae properties as a result of various fiber misalignment angles.