In large collections of tumor samples, it has been observed that sets of genes that are commonly involved in the same cancer pathways tend not to occur mutated together in the same patient. Such gene sets form mutually exclusive patterns of gene alterations in cancer genomic data. Computational approaches that detect mutually exclusive gene sets, rank and test candidate alteration patterns by rewarding the number of samples the pattern covers and by punishing its impurity, i.e., additional alterations that violate strict mutual exclusivity. However, the extant approaches do not account for possible observation errors. In practice, false negatives and especially false positives can severely bias evaluation and ranking of alteration patterns. To address these limitations, we develop a fully probabilistic, generative model of mutual exclusivity, explicitly taking coverage, impurity, as well as error rates into account, and devise efficient algorithms for parameter estimation and pattern ranking. Based on this model, we derive a statistical test of mutual exclusivity by comparing its likelihood to the null model that assumes independent gene alterations. Using extensive simulations, the new test is shown to be more powerful than a permutation test applied previously. When applied to detect mutual exclusivity patterns in glioblastoma and in pan-cancer data from twelve tumor types, we identify several significant patterns that are biologically relevant, most of which would not be detected by previous approaches. Our statistical modeling framework of mutual exclusivity provides increased flexibility and power to detect cancer pathways from genomic alteration data in the presence of noise. A summary of this paper appears in the proceedings of the RECOMB 2014 conference, April 2-5.