# On March Madness and perfection

Last week, several media outlets, and many well respected mathematicians, gave their best shot at estimating the odds of replicating a perfect March Madness bracket. Here’s are the two numbers I saw most often:

1 in 9.2 quintillion: This estimate assumes each game is coin flip.

1 in 128 billion: This calculation, estimated by a DePaul mathematics professor, operates under the assumption that game and team-specific information are used to make well-conceived picks.*

Unfortunately, while both of these estimates are valid, they belie the parameter which is of most interest, which is the following:

What are the odds that a random individual fills out a bracket with all 63 games picked correctly?

Why is this question of interest? Because nearly all of us filling out a March Madness brackets don’t flip a coin, and, perhaps just as egregiously, we don’t make well-conceived picks. As a result, estimates which are based upon either of those assumptions are borderline worthless in answering the question we are truly interested in.

Here’s an example:

In the 2014 first-round** contest between Mercer and Duke, 97% of the public picked the Blue Devils, according to ESPN. That’s certainly not a coin-flip, and worse, its not a well-conceived pick: stats-guru Ken Pomeroy gave Mercer at least a 15% chance of an upset. Assuming Pomeroy’s upset probability as the truth, more than 15% of sheets would have picked Mercer if they were drawing winners purely by chance. If that had been the case, almost 5 times as many sheets would’ve still been alive for perfection when compared to the public’s perception.

So, how can we answer the question of interest?

Well, to start, given that the tournament is not yet complete, let’s change the question to:

Q1: What are the odds that a random individual has filled out a bracket with all of the first 48 games picked correctly?

And I’ll add a bonus question:

Q2: What are the odds that a random individual has filled out a bracket with all of the first 48 games picked correctly, given that he or she randomly drew winners using statistical probabilities?

To answer Q1, I’ll operate under the assumption that the numbers posted on ESPN are accurate representations of everyone that fills out a bracket. Thus, I extracted the Sweet-16 probabilities for the remaining teams (16 of them, coincidentally), and the 16 Rd. 1 win probabilities from the teams that lost in Rd. 2 after winning their first game.

To answer Q2, I used Ken Pomeroy’s probabilities, posted on his blog before the tournament, for the identical teams and rounds.

Here’s a table of the teams which advanced exactly one round (and their Rd. 1 win probability), and the Sweet 16 teams (and their associated probabilities of making it at least that far).

It’s no surprise that the public hit on several favorites (including Michigan State) relative to Pomeroy, but the stats-guru had much higher Sweet-16 probabilities for Connecticut, Tennesee, Stanford, and Dayton.

For each prediction type (ESPN, Pomeroy), multiplying the numbers in the table above can give us estimated answer to Q1 and Q2:

Q1: What are the odds that a random individual has filled out a bracket with all of the first 48 games picked correctly?

`About 740 billion to 1.`

Q2: What are the odds that a random individual has filled out a bracket with all of the first 48 games picked correctly, given that he or she randomly drew winners using statistical probabilities?

`About 32 billion to 1.`

All together, through the first 48 games of 2014, picking games randomly using Pomeroy’s probabilities, relative to ESPN consensus, would have increased the probability of an individual selecting a perfect bracket by about a factor of 23!

A few final notes:

-The odds of a perfect bracket through 48 games, when using a coin flip for each game, are roughly 281 trillion to 1. Thus, ESPN has done about 380 times better than picking games by tossing a coin, and Pomeroy has done about 8800 times better.

-At tournament’s end, I’ll repeat the calculations using all 63 games. If Michigan St. or Florida wins, that would seem to help the ESPN consensus numbers, as those two teams were heavily backed. Other champions appear likely to support Pomeroy’s probabilities.

-If anyone has an archived version of 538’s win probability matrix, I’d enjoy using that as a comparison, too. The website currently shows updated probabilities only.

Footnotes:

*The article linked also quotes Minnesota biostatistics professor Brad Carlin. That’s the good news, as he is a star in the field. The bad news? USA Today writes “Carlin is professor of something called biostatistics.”

**Take that, NCAA: I’m still calling it the first-round. Play-in games don’t count.