## Monday, January 16, 2006

### Sports Droughts

We watched the Chicago Bears Football team lose to Carolina with some friends who were extremely pessimistic the entire game, even though the game remained close throughout. "We haven't seen the Bears win the Super Bowl in twenty years, they will continue to disappoint us."

Let's consider the twenty year statement. Let's assume that each year every team has an equal probability of winning and each year is independent of each other. Then the expected number of years between championships is equal to the number of teams, 32 in the National Football League. So the Bears are still ahead of the curve, not disappointing at all.

So how about the 86 years between the Boston Red Sox World Series championships in baseball, the 88 years between Chicago White Sox championships, and the 98 years since the Chicago Cubs last won? This is just the coupon collector problem where if one draws numbers 1 to n with replacement independently and uniformly, it will take an expected n ln n draws to see every number. For the thirty baseball teams, that makes 102 years. There was no curse for the Red Sox, White Sox and Cubs, just probability working as expected.

I'm cheating on many fronts. The number of teams in both football and baseball have grown dramatically over the past few decades. Each year is not independent; a good team one year will likely be good the following year. Teams do not have an equal probability; especially in baseball the richer teams have a higher chance of winning.

Nevertheless you have no one to blame for long losing streaks other than those evil gods of probability.

1. Of course, probability doesn't explain teams that come heartbreakingly close, again and again, only to lose each time (see: Red Sox, pre-2004) =).

2. "Nevertheless you have no one to blame for long losing streaks other than those evil gods of probability."

This is nonsensical. What about the Yankees having won about one third of the World Series over the past 80 years?

3. I wonder - if you collect O(n) coupons and then sort according to occurrance, what the histogram will be like? Will there be a short plateau of "winners" in the beginning?

4. 2nd Anon, I don't see how your question is relevant to Lance's post. In the case of unequal probabilities, losing streaks only get longer...

5. Yes, so losing streaks have much more to do with the third baseman, pitching staff, general manager, etc. than with the "evil gods of probability." There are plenty of people to blame...

6. For baseball there is an argument for a bias towards even longer droughts than predicted by the coupon collector problem, but in the NFL the biased schedule more often pits the previous year's weak teams against each other and strong teams against each other. This (together with salary caps) should reduce the length of droughts, not increase them.