Problem of the Week 973

Penalty Kick Coincidence

This problem is dedicated to the surprising Macalester men's soccer team, who advanced to the national quarter-finals on the basis of two exciting penalty kick shootouts, won by scores of 3-2 and 3-1. Thanks for a great season from a devoted fan. (They lost to Trinity (Texas) 0-1 in the quarter-final match.)

If the chance of success for a penalty kick is taken to be 50%, what is the most likely score in a penalty kick shootout?

Note: A penalty kick shootout in soccer has five rounds of the form: Team A shoots; then team B shoots. If at any point the score is such that a team is guaranteed to win, the shootout ends. In case of a tie after five rounds, additional rounds are played until there is a winner. A score is reported as m-to-n where m > n (regardless of who shot first).

© Copyright 2002 Stan Wagon. Reproduced with permission.

3 December 2002