# Double spend races

## TL;DR

This paper refines the analysis of double spend attacks in Bitcoin, providing a precise probability formula, correcting previous assumptions, and offering a more detailed risk assessment based on confirmation counts and validation times.

## Contribution

It corrects Nakamoto's original double spend analysis, introduces a closed-form probability formula, and offers an asymptotic approximation for better risk evaluation.

## Key findings

- Probability of double spend success decreases exponentially with confirmations.
- Larger confirmation counts are needed than previously estimated.
- Conditional probability analysis improves risk assessment accuracy.

## Abstract

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.