Abstract

Sporting events generate many interesting probability questions. In probability, counterintuitive results are common and even experienced mathematicians can make mistakes in problems that sound simple on the surface. In this article we examine two sports applications in which probability questions have results that may come as a surprise to sports fans and mathematicians alike. The first concerns the chances of rare events in sporting contests, with recent examples from baseball and football. The second considers the merits of longer playoff series in sports such as major league baseball and professional basketball, where a set of games is used to determine who advances in a post-season tournament.

Problem 1: Rare Events

One of the most appealing aspects of watching sports events is the possibility that a viewer may see something especially noteworthy, perhaps even unprecedented. Sports fans who have the good fortune to be in attendance when something spectacular happens have vivid memories and wonderful stories. Baseball fans love to describe the time they saw a no-hitter, a player hit for the cycle, a triple play, or some other rare event. A typical response to such descriptions is, “Wow, what are the chances of that?”

Those of us who love sports and math frequently are asked, “What are the chances of …?” It may be easy to come up with a quick estimate of the probability, but first guesses are often wrong, sometimes off by orders of magnitude! Making estimates of the probability of rare events can be difficult.