# Can Anyone Win the Billion Dollar Bracket?

About five years ago, Brian Burke – of Advanced NFL Stats – wrote a post in this forum on modeling win probability in college basketball, demonstrating that Brian Burke can do more than just talk football and stats!  Then this past week, Dave Collins – host of the Advanced NFL Stats podcast (where he interviews football writers, analysts, and researchers) – contacted me about more college basketball related research from the Advanced NFL Stats people.  The specific research comes from Brad Null and bracketvoodoo.com. What follows is Brad’s comment on the latest wrinkle to March Madness: The Billion Dollar Bracket.

We’ve all seen the odds of picking a perfect bracket at random—effectively the same thing as flipping heads 63 times in a row or 1 in 263 (or 1 in 9,223,372,036,854,775,808 if you’re not into the whole brevity thing). But you could improve these odds considerably by picking all of the favorites, and using 2013 data, I analyzed that doing so would give you a 1 in 80 billion chance of picking a perfect bracket. Looking at an actual projected bracket for this year however, and given the lack of clear separation among the top 16 or so teams this year, these odds are actually closer to 355 billion to one. Let me break down how we come to that number.

Using bracketvoodoo.com’s proprietary NCAA Tournament prediction system, we calculated the probability of the favorite winning 63 consecutive games in Jerry Palm’s most recent projected bracket (see our picks here). The favorites for the 32 projected Opening Round games have a win probability ranging from 51% to 98%. Multiplying these together we get .0019%. That doesn’t sound that bad; that’s a 1 in 53,000 chance of getting the whole first round correct.

If we assumed the remaining 31 picks were about as challenging as the first 32, then we would expect the odds of this bracket coming out perfect to be 1 in 53,000^2 or about 1 in 2.8 billion. Unfortunately, though, it is actually significantly harder to predict the last 5 rounds than the first one. This is because the games at that point are expected to be much more competitive, especially if all the favorites were to win in the first round. Assuming this bracket was perfect through the first 32 games, and we continued with this strategy of picking all favorites, our probability of getting each of the next 31 picks correct would vary between 52% and 78%, and our chances of getting them all correct would be .000015% or 1 in 6.7 million. This is 125 times harder than picking the entire first round correctly. And this is why the best odds you can give yourself of correctly predicting the entire NCAA Tournament bracket are around 1 in 355 billion.

To look at it another way, the geometric mean of the 32 first-round probabilities is 71% while the geometric mean of the 31 subsequent probabilities is 60%. This means that the best-case chance of picking the first round correctly is equivalent to .71^32, and then subsequently picking the rest of the games correctly would be equivalent to .60^31, which is a much smaller number.

So having made the argument that YOU will not win a billion dollars, what then are the odds that anyone will win it, given that up to 15 million people will have a shot at it? An easy upper bound is 15 million divided by 355 billion or 1 in 24,000. In reality though, there are a few things we don’t know. 1) Will 15 million people enter this contest, 2) how likely will the brackets they enter be, and 3) how many duplicates will there be?

Let’s assume at this point that 15 million people do enter. Leveraging the historical analysis that we have done on bracket picking behavior in the past, we estimate that the average bracket picked for such a contest would be about 20x less likely than the “all-favorites bracket” described above. And while there will be some duplicate entries, our analysis indicates that this would eliminate less than 10% of all brackets. Thus, there would be at least 13.5 million unique brackets among this set. Put that all together and you get about a 1 in 500,000 chance that Warren Buffett has to shell out that billion dollars. So the EV on that insurance policy he wrote is about \$2000. I’m guessing he got a pretty nifty premium though.