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The supercell-based plane wave expansion method is used to study the effects of random disorders on the defect states of a two-dimensional solid-solid phononic crystal with a point defect. Phononic systems with plumbum scatterers embedded in an epoxy matrix are calculated in detail. The radius disorder and location disorder are concerned. The influences of the disorder degree on the defect states for both anti-plane and in-plane wave modes are investigated and discussed. It is found that, with increase of the disorder degree, the frequencies and localization degree of the defect bands will change. The influence of the disorder on the defect states is more pronounced for the in-plane modes than for the anti-plane modes. Radius disorder has more influences on the defect bands than the location disorder does. The analysis of this paper is relevant to the assessment of the influences of manufacture errors on behaviors of elastic wave propagating in wave filters, as well as the tuning of the point defect states.