This dissertation proposes a model of phonological relationships, the Probabilistic Phonological Relationship Model (PPRM), that quantifies how predictably distributed two sounds in a relationship are. It builds on a core premise of traditional phonological analysis, that the ability to define phonological relationships such as contrast and allophony is crucial to the determination of phonological patterns in language. The PPRM starts with one of the long-standing tools for determining phonological relationships, the notion of predictability of distribution. Building on insights from probability and information theory, the model provides a way of calculating the precise degree to which two sounds are predictably distributed, rather than maintaining the traditional binary distinction between "predictable" and "not predictable." It includes a measure of the probability of each member of a pair in each environment they occur in, the uncertainty (entropy) of the choice between the members of the pair in each environment, and the overall uncertainty of choice between the members of the pair in a language. These numbers provide a way to formally describe and compare relationships that have heretofore been treated as exceptions, ignored, relegated to alternative grammars, or otherwise seen as problematic for traditional descriptions of phonology. The PPRM provides a way for what have been labelled "marginal contrasts," "quasi-allophones,""semi-phonemes," and the like to be integrated into the phonological system: There are phonological relationships that are neither entirely predictable nor entirely unpredictable, but rather belong somewhere in between these two extremes. The model, being based on entropy, which can be used to understand the cognitive function of uncertainty, provides insight into a number of phenomena in synchronic phonological patterning, diachronic phonological change, language acquisition, and language processing. Examples of how the model can be applied are provided for two languages, Japanese and German, using large-scale corpora to calculate the predictability of distribution of various pairs of sounds. An example of how empirical evidence for one of the predictions of the model, that entropy and perceptual distinctness are inversely related to each other, could be obtained is also provided. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]