Monday, December 24, 2007

What the Patriots Can Teach Us About Futures Odds

A coin has landed on heads 15 times in a row. What are the chances of it landing heads on the 16th flip?

Many of you will pick one of three answers:

- The coin is "hot", so the chances are greater than 50%.
- The coin is fair, so the chances are exactly 50%.
- The coin is "due", so the chances are less than 50%.

These answers are all using incorrect logic. The correct answer considers the possibility of a loaded coin. Loaded coins are rare enough that when we start flipping, we don't think about them. But when 15 consecutive heads are flipped, Bayesian probability tells us that the chances of a loaded coin have gone way up. Thus, the probability of a heads on the 16th flip is significantly greater than 50%.

Now think about analyzing the probability of a team going 16-0 before the season starts. If you're a good handicapper, you can estimate the moneyline for each game and multiply the team's winning probabilities together. But this wouldn't give you an accurate estimate.

If a team is 15-0, chances are very good that the team is much better than you believed at the start of the season. Certainly this is the case for the 2007 Patriots; their season wins over/under was set at 11.5. Thus, your preseason estimate for the 16th game is likely to underestimate their chances of winning. The same is true for other intermediate games. So if you really want to handicap the chances of an undefeated team, or a similar proposition, you must account for the probabilities of teams performing much better or worse than you expected.

This principle has other applications as well. Many preseason baseball simulations have 80-projected-wins teams almost never making the playoffs (except in the NL Central, of course). But 80 is just a mean estimate; there's a significant chance the team actually has 88-win talent. A team that has a fixed talent level of 80 wins will make the playoffs very rarely, but if their actual talent follows a normal distribution from 70 to 90 wins, they have a legitimate shot. (Remember, a 90-win team can still overperform and win 100 games.)

And that, kids, is how the Colorado Rockies made it to the World Series. Colorado probably won't repeat, but I'd bet that at least one ".500 team" will play in October next year.

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