Mile Run. Given data on men's and women's records for the mile run during the twentieth century, project the course of future records. Is it likely that a woman will ever break the men's record?
Tie Breaker. Given the complete record of a soccer league that ended in a three way tie, devise a fair means of determining who should be crowned champion.
Playoff Series. Use simulation, backed up by probability calculations, to see how the length of a playoff series affects the chances of a weaker team pulling off an upset by winning the series.
Fair Competition. Consider a simple game in which two players alternate tossing two die. Player A gets a point if the product is even; player B gets a point if the product is odd. Is this game fair? If not, what change in the rules or in the scoring system might be made to make it fair?