Abstract
The "Big Five" European football leagues, comprising England's Premier League, Germany's Bundesliga, Spain's La Liga, Italy's Serie A, and France's Ligue 1, command significant attention. While the occurrence of goals, substitutions, and fouls in football games is often considered random, of the presence of an inherent inevitability is unclear. To investigate, we analyzed a public dataset detailing timing of goals, substitutions, and yellow cards in regular time from WhoScored across three seasons (2018-2019, 2019-2020, 2020-2021) in the top five European football leagues. We employed various mathematical descriptive models (including linear, sigmoid, and gaussian functions) to measure the temporal tendency of goals, substitutions, and yellow cards. Our results indicate that, whether in the first or second half of the match, the temporal distribution of these elements exhibits evenness a (indicative of randomness). However, specific characteristics were discerned through distinct model parameters, capturing novel phenomena that were intuitively illustrated. Furthermore, we explored the interaction of the timing of goals, substitutions, and yellow cards. In this analysis we found that scoring in the second half leads to more substitutions and yellow cards. Changing players in the second half corresponded with more goals, while the impact of yellow card fouls showed no differences in goals in the first and second halves. Our research is the first to systematically study the laws of modern football matches, providing valuable guidance and reference for many football coaches.