Hoops

With the NCAA tournament about to tip off today, I thought it would be the perfect opportunity to discuss an exceedingly relevant topic to basketball and tournaments: conditional probability and Bayes theorem.

Yes, I may be taking extreme ironic pleasure in mentioning basketball players in the same sentence as mathematical concepts, but whether regardless of whether our NCAA athletes can in fact even pass a math test does not somehow exclude them from the grasp of statistics.

So what the hell is the connection between probabilities and tournament performance? I’m quite sure you’ll regret that question in just a moment as we dive deep into concepts that only Nobel Laureates and bespectacled academics get worked up for.

Let’s start with a somewhat simple example. Let’s say we’ve flipped a coin ten times, and we’ve somehow managed to come up with heads for each outcome. If we flip the coin again, what are the odds that we’ll come up with heads yet again? This isn’t a trick question: your powers of deduction haven’t failed you, assuming that you have correctly answered 50 percent.

And why? Despite our incredible desire to somehow believe that the universe craves balance and must intervene on behalf of our fate-tempting coin, we’ve proven to the best of our abilities that this is not the case. Each flip is independent, and despite the incredible unlikelihood of flipping 10 heads in a row – over a one in a thousand chance – there is a fifty percent likelihood that the streak will continue, or bust. As much as we would like to hope and believe in some sort of fate, in an invisible hand guiding us, it all comes down to chance.

Many things in life, though, do not share this property of independence. In fact, most of the events we experience are heavily conditioned on events that have occurred before; fate rarely strikes us across the forehead with chance. Your grades in a class are largely influenced by the grades you receive in all your classes prior. You going to a particular school or getting a job in a particular city are all based heavily on the experiences that have shaped you up to that point. Conditional probability and Bayes rule both boil down to a rather simple adage:

Past performance is a good predictor of future results.

This of course, is the link to the March Madness tournament. Preseason rankings, margins of victory, injuries, even where the games are being held all influence the expected outcomes of the games. If you’re interested in one such model of all these factors, check out Nate Silver’s bracket, and his more detailed explanation of the pick process. And yes, if you’re wondering, that is the same Nate Silver who correctly predicted every single major election outcome last November. Sadly, basketball productions are harder to reliably divine.

Where conditional probability really starts to throw its weight around is when the tournament actually starts. A 16-seed has a statistical impossibility to win the tournament when the field is 64, but if you already know that they’ve made it to the Final Four, the odds are drastically different. They’ll still undoubtedly be the underdog, but the odds are much more manageable.

Does any of this really help you make last-minute picks? Not in the slightest. But it is a reminder that math can sometimes be interesting, informative, and influential.