From Expected Goals to Scoring at Least Once: An Event-Specific Summary of Aggregated Bernoulli Risk

Expected goals (xG) is widely used to quantify offensive performance in football by summarizing the expected number of goals from shot-level scoring probabilities. However, xG reflects only the first moment of the underlying Bernoulli system and does not capture how scoring probability is distributed across shots. As a result, teams with identical total xG may nevertheless have different probabilities of scoring at least once. In this paper, we study the quantity $x{G}^{+}=-\log \mathbb{P}(G=0)$, which is a monotone transform of the exact no-goal probability and, equivalently, of the probability of scoring at least once. We interpret $x{G}^{+}$ as an additive, event-specific summary of aggregated Bernoulli risk and analyze its main structural properties. In particular, we show that $x{G}^{+}\ge xG$, with equality only in the degenerate case $p_i=0$ for all i, and we derive a second-order approximation linking $x{G}^{+}-xG$ to the second moment of shot probabilities, the effective number of shots, and Rényi-2 entropy. Empirical illustrations on football data show how concentrated shot profiles can increase scoring certainty relative to total xG and how exact Bernoulli aggregation differs from a Poisson approximation based only on the mean. While xG remains an appropriate measure of expected scoring volume, $x{G}^{+}$ provides a complementary summary targeted at the probability of scoring at least once.