Uses and limitations of mathematics in sport

A number of topics are reviewed under the theme of the interaction between mathematics and sport, mainly based on probabilistic or statistical considerations. Some differences and similarities that arise from different scoring systems, exemplified by motor racing and table tennis, are described. Ways of resolving ties in league tables, or drawn matches in individual contests, are explored, with examples from chess and soccer: a suggestion is made for modifying the usual penalty shoot-out in soccer so as to minimize its expected length. Two different ways of assessing the ‘importance’ of single incidents within a larger contest are offered: both suggest that such incidents are equally important to both contestants. Sample calculations indicate that the importance of incidents may be fairly constant across a wide range of values for the parameters that govern the outcome of a contest. Two statistical phenomena—Simpson's paradox and regression to the mean—are illustrated by basketball and golf. Applications of dynamic programming, to American football and soccer, are outlined. Different ways of computing batting averages in cricket, perhaps more meaningful than those generally used, are noted.