Lightning photo courtesy of cnx.org.

The NCAA tournament has become a cultural phenomenon where everyone suddenly becomes a college basketball expert whether or not we’ve ever watched a game. This expertise has raised to a fever pitch this year as Quicken Loans has offered 1 billion dollars to anyone who provides a perfect bracket. But why?

This contest, underwritten by Warren Buffett’s Berkshire Hathaway, is often described as a one-in-9 quintillion change of winning, or 2^63rd power based on there being 63 games played by 64 teams in this single-elimination tournament. However, this model is obviously wrong since we know that some of these teams are better than others. Even the most casual NCAA bracket filler knows that a 1-seed (presumably one of the top 4 teams in the country) always beats a 16-seed (which is typically one of the worst 4 teams in the tournament). Similarly, a 2-seed almost always defeats a 15-seed with rare exceptions. After this point, the expectations start get a little trickier and the March Madness descends into full effect.

Even so, we know that top seeds tend to be safe well into the second week of the tournament. So, given that the NCAA games are not a pure coin toss, what are the real odds of filling out a bracket perfectly?

There are a few ways to go about this estimate. One is to go through historical data and look at how the NCAA bracket has carried out over time, then use the odds that each seed moves to the next round as the basis of a win expectancy at each round. This is the approach that the quant in me wants to take and it is definitely the most tempting way to go. However, a couple of basic problems keep me from taking that approach.

First, this approach lacks the specific context of how college basketball exists in 2014. Comparing NCAA tournament results from 50 years ago, when freshmen weren’t allowed to play college basketball and John Wooden’s UCLA Bruins owned the tournament to today’s world where “one-and-done” freshmen are often the best players on their team and talent is more distributed throughout the country seems to be an unfair comparison. Also, seeds are determined very differently, with power rankings, automatic berths, and decision makers changing on a year-to-year basis. The tournament today is very different today than it was even 10 years ago. In addition, we know that there are specific biases in seeding that seem to be errors, such as Louisville’s seeding as a 4-seed even as many experts believe that they are a top-4 team. These individualized biases make a deep longitudinal study an interesting historical exercise, but not necessarily the best predictive model.

Second, and more importantly, I’m on a plane right now and don’t have access to the numbers.

So, in the lack of true quantitative evidence, I created a simple qualitative model where I estimated the ability to choose the correct winner of each game. We can assume that almost everybody will bet on the 1 seeds to win the first round and be correct, so this initial assumption changes the odds from 1 in 9 quintillion to about 1 in 600 quadrillion. (Quicken Loans may as well just hand the money away right now…)

In my model, 1-16 game was a 99% chance of choosing the winner, a 4-13 game was treated as an 80% change of winning, while an 8-9 game or a semifinal or final game was treated as a 55% chance of picking the winner, since there’s almost always some level of information that shows that there is a favorite. Basically, as the talent differential gets smaller, the picks are increasingly due to chance. Put that together and the odds start changing significantly.

When I did this back-of-the-envelope calculation, I came up with much lower odds of 1 in 298 billion to get a perfect bracket. These odds are still astronomically high, but start to get closer to the real number. I’m sure that Warren Buffett went through a similar process, discounted this number significantly, then provided his insurance policy accordingly.

But this model has a flaw (now that I’m back on the ground in Orlando). It assumes that you will always choose the higher seed, whereas this isn’t always the case. Ed Feng of Grantland and Stanford identified a much better model that takes into account both the potential outcome that either the higher or lower seed would win. With his model, the odds are 1 in 4.5 billion. That’s still a really high number. In contrast, your odds of being killed by lightning are 1 in 280,000 (http://www.ehow.com/info_8607019_chances-being-hit-lightning.html). Yes, you’re 16,000 times more likely to be killed by lightning than to win.

So, how much would this policy for a billion dollars actually be? If you’ve got 15 million players for the billion dollar bracket, you’d expect a 1 in 300 chance to win. So, the back of an envelope approach says that you could expect to come out ahead by providing an insurance policy for a bit over $3 million.

You can start to play with the numbers a bit more to be more conservative, but it’s hard to price the maximum breakeven at much more than $5 million based on reasonable assumptions. A gut feeling says that Berkshire Hathaway probably felt comfortable with charging $5-$10 million as a policy for Quicken Loans. Based on Buffett’s philanthropic nature, let’s call it $5 million.

Now, for the next part. Does this make sense for Quicken Loans? According to Dan Gilbert, Quicken expects to get 15 million new leads from this process, meaning 15 million new potential customers for mortgages and other loans. Again, playing the back of the envelope card, assume that Quicken gets an average loan of $150,000 from each closed deal. Based on a 30 year loan, 6% interest and 3% inflation, Quicken gets about $98,000 in discounted interest. Add closing costs and let’s just say Quicken makes $100,000 per loan.

So, to break even on the insurance policy, all Quicken really needs to do is find 50 mortgages out of all of this. Can Quicken Loans convert 1 in every 300,000 qualified contacts into a mortgage? Probably so, based on their brand name and the assumption that their sales force knows how to qualify and close interested parties.

But in this context, all of the marketing around the billion dollar bracket suddenly makes more sense. Even if you include the marketing costs and all of the other effort that Quicken is putting into this, the end result is that they are getting millions of people’s verified contact information for what ends up being a small fraction of their potential value.

Now that you know the real numbers behind the Quicken bracket, the story changes considerably. The real story isn’t “Can you win a billion dollars based on a 1 in 9 quintillion chance of winning?” The real question is “Can Quicken Loans get, say, 250 new mortgages out of their marketing campaign to justify the marketing and insurance efforts they’ve put in place?”

And at the end of the day, everybody wins. Warren Buffett makes another 5 million dollars. Quicken Loans probably makes 50 million dollars. Yahoo gets its marketing money. And we all continue to get an online platform that helps us to continue our crack-like addiction with March Madness. Nobody loses. (Unless you’re a competing mortgage provider.)