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Double spend races

We correct the double spend race analysis given in Nakamoto's foundational Bitcoin article and give a closed-form formula for the probability of success of a double spend attack using the Regularized Incomplete Beta Function. We give a proof of the exponential decay on the number of confirmations, often cited in the literature, and find an asymptotic formula. Larger number of confirmations are necessary compared to those given by Nakamoto. We also compute the probability conditional to the known validation time of the blocks. This provides a finer risk analysis than the classical one.