This paper examines the fairness of recidivism prediction instruments (RPIs) and their potential for disparate impact. It discusses several fairness criteria, including calibration, predictive parity, error rate balance, and statistical parity, and demonstrates that these criteria cannot all be satisfied simultaneously when recidivism prevalence differs across groups. The paper shows that disparate impact can arise when a RPI fails to satisfy the criterion of error rate balance, leading to stricter penalties for high-risk individuals in one group compared to another. The analysis is supported by empirical results based on data from Broward County, Florida, and highlights the importance of considering the distributional differences in scores between groups. The paper concludes by discussing the implications of these findings for the use of RPIs in criminal justice decision-making and the need to ensure that such instruments are free from biases that could lead to disparate impact.This paper examines the fairness of recidivism prediction instruments (RPIs) and their potential for disparate impact. It discusses several fairness criteria, including calibration, predictive parity, error rate balance, and statistical parity, and demonstrates that these criteria cannot all be satisfied simultaneously when recidivism prevalence differs across groups. The paper shows that disparate impact can arise when a RPI fails to satisfy the criterion of error rate balance, leading to stricter penalties for high-risk individuals in one group compared to another. The analysis is supported by empirical results based on data from Broward County, Florida, and highlights the importance of considering the distributional differences in scores between groups. The paper concludes by discussing the implications of these findings for the use of RPIs in criminal justice decision-making and the need to ensure that such instruments are free from biases that could lead to disparate impact.