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A method is proposed for computing with unreliable nanoscale devices that have a high rate of transient errors. Errors are corrected using a probabilistic circuit in which device noise is leveraged as a computational asset. Example designs that achieve a low output bit error probability are presented. The effect of permanent defects is also evaluated, and transient device noise is found to be beneficial for correcting hard defects for defect rates of as high as 0.1% and transient fault rates above 1%. When compared with existing fault-tolerant methods, the sample design requires considerably fewer redundant gates to achieve reliable operation. These results predict that some degree of engineered randomness may prove to be a useful signal-processing feature in future nanoelectronic systems.