  We investigate the time evolution of lead changes within individual games of
competitive team sports. Exploiting ideas from the theory of random walks, the
number of lead changes within a single game follows a Gaussian distribution. We
show that the probability that the last lead change and the time of the largest
lead size are governed by the same arcsine law, a bimodal distribution that
diverges at the start and at the end of the game. We also determine the
probability that a given lead is "safe" as a function of its size $L$ and game
time $t$. Our predictions generally agree with comprehensive data on more than
1.25 million scoring events in roughly 40,000 games across four professional or
semi-professional team sports, and are more accurate than popular heuristics
currently used in sports analytics.
