Original Post as of March 18, 2014. Reposted with minimal changes on March 16, 2016. Some of the references (such as the $1 billion challenge) will not apply, but the overall logic still does.
Warren Buffet made quite the buzz when he announced that he would award $1 billion to anyone who could fill out a perfect NCAA bracket. We have all seen the number being thrown around over the last several days. “Your odds of filling out a perfect bracket are one in 9.2 quintillion, or 9,223,372,036,854,775,808.” The problem with this number is that, while technically mathematically accurate, it doesn’t really give an accurate portrayal of one’s chances. The odds are far better, but I won’t mislead you into thinking that anyone can fill out a perfect bracket. No one has ever done it so far, and no one will likely ever do so (without the help of a computer program that creates billions of entries). So how “great” are the odds then? Let’s dive into the numbers.
The 9.2 Quintillion Number
So where did the 9.2 quintillion number come from and why is it off? The number came from 2 to the 63rd power, which is 2 times 2 a whopping 63 times. That gives us our outlandishly large number. The problem with this number is that it operates under the assumption that in every game, there is a 50/50 chance of either team winning. Anyone that knows anything about basketball, or even just March Madness, knows that this is not the case. While upsets happen, the higher seeds win the majority of the time. So clearly this number is not an accurate representation, and the odds are much higher.
The Numbers Game
We have established that a 50% chance of winning per game is wrong, but how can we accurately predict such an unknown quantity? Luckily someone far smarter than me figured out a better system. Jeff Bergen, a math professor at DePaul University, came up with a system that was very impressive and fairly accurate. His system estimated the chances that higher seeds would win out, thus giving the relative probabilities of an educated fan selecting a perfect bracket. His conclusion was that the odds were 1:128 billion. The only issue that I had with his methods is that he used some probability assumptions (albeit logical ones) instead of historical data. I am using his methodology, but plugging in historical data. All the data used was taken from Mcubed.net. A full spreadsheet showing the advanced calculations that I used can be found in this Google Spreadsheet.
|Round of 64 (Round 2)||Odds of Higher Seed Win||Odds (Approx)|
|1 vs. 16||100.0%|
|2 vs. 15||94.0%|
|3 vs. 14||85.3%|
|4 vs. 13||78.4%|
|5 vs. 12||67.6%|
|6 vs. 11||66.9%|
|7 vs. 10||60.0%|
|8 vs. 9||51.4%|
|Perfect Region (To this point)||8.8%||1 in 11|
|Perfect Bracket (To this point)||0.0059%||1 in 17,000|
For the purposes of this article, I will refer to the rounds by their official name. The play-in games are treated as the first round, so the round of 64 is technically the second round. After the second round, your chances of having a perfect bracket have already plummeted to .0059%, or about one in 17,000. With the number of brackets submitted each year, there will probably be a small handful this year that are still perfect after round two.
|Round of 32 (Round 3)||Odds of Higher Seed Win||Odds (Approx)|
|1 vs. 8||81.1%|
|2 vs. 7||74.4%|
|3 vs. 6||54.3%|
|4 vs. 5||55.4%|
|Chance of Perfect Region||1.59%||1 in 63|
|Chance of Perfect Bracket||0.000006414%||1 in 15.6 million|
The gap from the second and third rounds is extreme, and almost no brackets will survive. The numbers say that the chances of remaining perfect are about 1 in 15.6 million. The good news is that you still have a better than 1% chance of having a perfect region at this point, so you’ve got that going for you.
|Round of 16 (Round 4)||Odds of Higher Seed Win||Odds (Approx)|
|1 vs. 4||67.2%|
|2 vs. 3||61.0%|
|Chance of Perfect Region||0.652%||1 in 153|
|Chance of Perfect Bracket||0.000000181%||1 in 550 million|
I don’t know if anyone’s bracket has ever made it this far, but it seems quite unlikely that anyone’s has. The odds jump to 1 in 550 million, which are worse than the odds of winning the Mega Million Lottery (1 in 258.9 million).
Rounds Five, Six, and Seven
|Round of 8 (Round 5)||Odds of Higher Seed Win||Odds (approx)|
|1 vs. 2||54.5%|
|Chance of Perfect Region||0.3555323512286200%||1 in 281|
|Chance of Perfect Bracket||0.00000001598%||1 in 6.25 billion|
|Round of 4 (Round 6)||Odds of Higher Seed Win||Odds (Approx)|
|1 vs. 1||50.0%|
|Chance of Perfect Bracket||0.0000000040%||1 in 25 billion|
|Championship (Round 7)||Odds of Higher Seed Win||Odds (Approx)|
|1 vs. 1||50.0%|
|Chance of Perfect Bracket||0.00000000200%||1 in 50 billion|
It’s almost not even worth discussing the possibilities of making through these rounds, but I will summarize just to show how unlikely it is. At this point in the tournament, it is assumed that each game is almost a 50/50 chance. While this seems favorable, imagine having made it to this point unscathed, then flipping 8 coins and needing all of them to land on heads. Not likely.
So what is the final number, adjusted for basketball knowledge and historical data? One in 50 billion. While this is still highly unlikely, it is far better than the 9.2 quintillion number, or even the 128 billion number proposed by Professor Bergen. The entry limit for the Quicken $1 Billion Contest is capped at 15 million. Even if you maxed out your entries, you would still have a 1 in 3,333 chance of winning. If that is worth it to you, then get to filling out those brackets. Enjoy the tourney and let the Madness ensue.