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Friday, March 19, 2010

The Perfect Bracket

What are your chances of getting a perfect bracket? Strictly speaking, the probability of getting every pick right is the same as flipping 63 straight heads, or 1 in 9,223,372,036,854,775,808. But that assumes that you are making picks without considering any other information-tossing darts at a dart board. Even using team mascots should improve your chances a little.

In reality, calculating the probability that someone creates the perfect bracket is a deep philosophical question. In theory, or theology, an omniscient deity should have a perfect bracket every time. But the theory of infinite complexity suggests that, for those of us bound by the laws of physics, approaching that kind of accuracy is impossible. The outcome of a basketball game is dependent on millions of millions of billions of different variables-e.g. what was the condition of the chicken that produced the egg that the team trainer, responsible for taping up the 9th man in the rotation, ate for breakfast and, more importantly, how did he cook said egg? In fact, there is an infinite number of variables that one would have to consider. It's physically impossible to measure, record, and model an infinite number of variables. Instead, we have to focus on a few, and then pray for the best, hoping that omniscient deity is also a basketball fan.

I decided to focus on just one variable-seed. What is the probability that the higher seed will win every game. We are, of course, reduced to coin tosses in the Final Four. The variable is easily measured-each team is explicitly labeled with a seed. We can also easily record past results to help us estimate probabilities. Modeling can get a little more complicated. The easiest solution is to use past results by round to estimate probabilities-e.g. all time, 1 seeds have beat 16 seeds every time, so we would estimate that the probability of a 1 beating a 16 to be 100/100=1. But in so doing, we would be throwing out information that we could use-e.g. 9 seeds have won 54% of games against 8 seeds, but 8 seeds are much more likely to beat 1 seeds, evidence that 8 seeds, on average, are better teams.

Using the first method, your chances of having a perfect bracket if you use only information on seeds comes out to 1 in 52,442,351,296. With the second method, you've got a 1 in 46,150,704,307 chance of getting it perfect. In other words, using seeds drastically improves your chances from zero to a slightly larger zero. But this is the average probability over the last 25 years. It is an estimate. In reality, these values could actually be smaller or larger, and the probability for any one season can be substantially higher or lower depending on how well teams are seeded and the distance between high and low seeds.

How much could we improve our chances using more variables? In theory, the best we mortals could do is to always pick the team favored by some information aggregating institution-the Vegas line to name one. If the line accurately reflects the median point differential, we can estimate the probability that team A beats team B by adding an estimate of the standard deviation. From that, we can estimate the probability that you would pick every game correctly. It turns out, this season is going to be especially difficult to predict (and the first day has made that more than apparent). Roughly, I have estimated the probability of a perfect bracket using Vegas odds at about 1 in 200 billion (1 in 188,642,197,125)-significantly better than 1 in 9 quintillion, but lower than the probability you would face in most seasons using seeds alone.

1 comment:

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