Unlocking the combat equation: do attackers or defenders have an extra advantage?

Closed-form combat equations, such as the Lanchester and Helmbold relations, have long been used to describe battle outcomes at an aggregate level, but they do not directly provide probabilities of victory or expected combat duration. Building on this tradition, this paper introduces a probabilistic combat equation that predicts both quantities within a symmetric modeling framework for attackers and defenders. The model is evaluated using historical combat data and an empirical probability formulation, with results expressed as functions of the fractional exchange ratio. A systematic comparison of attacker and defender perspectives reveals a statistically significant asymmetry in how the two views diverge from empirical probabilities, even though the underlying model contains no built-in attacker advantage. This divergence varies nonlinearly with the fractional exchange ratio and remains robust across alternative linear and quadratic specifications, including a localized analysis near unity exchange. Importantly, the implied attacker advantage does not necessarily indicate intrinsic operational superiority. Rather, it may reflect asymmetries in the historical record, including unbalanced classification of battle outcomes, differing interpretations of attacker and defender victories, and a concentration of observations in exchange ratio regimes favorable to attackers. The results highlight both the strengths and limitations of probabilistic combat models when applied to historical data.