## Tuesday, August 2, 2011

### Does Experience Matter?

The NFL is coming back, and many free agent veterans are on the move.  Sports talking heads often drone on about how much "experience matters" when it comes to sports.  There may be some logic to this—if you're having heart surgery, you certainly want to know your doctor's used a scalpel before—but it shouldn't make you feel good about your team overpaying some over the hill interception-prone QB just because he happened to win for a Super Bowl 14 years ago [See: Favre, Brett].  But if experience does matter, one should be able to see this statistically.  Is there a strong correlation between previous Super Bowl wins and future Super Bowl success?

According to ESPN The Magazine, the most experienced team has won the Superbowl 58% (25/43) of the time.  That's certainly more than half, but this could be due to random chance. After all, a 58% winning percentage over a 16 game season amounts to about 9 wins, which an average team could certainly achieve just by being lucky.  Assuming it is due to random chance, what's the probability that the more experienced team won 58% of the time?  To solve this, we need the binomial distribution  f (k; n, p) ,

where n = 43  is the number of trials, k = 25 is the number of success, and p = 0.5 is the probability of winning if the results were random.  The binomial coefficient (i.e. the weird "n over k" thing in the parentheses) is defined as

The function f (k; n, p) is the probability that a random event will produce exactly k success in n trials.  In this case, we want to know the probability that experienced teams have won at least 25 times in 43 trials.  Using Wolfram Alpha to sum up these probabilities, we find that there's an 18% chance of the more experienced team winning at least this many games.

In the last Superbowl, 29 former Super Bowl champs played for the Steelers and no former champs played for the Packers.  Clearly, it's not all about experience.