March Madness: What are the Odds of a Perfect Bracket?

What are the odds of a perfect bracket? saje/saje
What are the odds of a perfect bracket?

March Madness is here!  Last year, Warren Buffett, Quicken Loans, and Yahoo! made headlines with the billion-dollar contest for a perfect NCAA Men’s Basketball bracket.  This year, unfortunately, we won't have the billion-dollar prize out there, because of a lawsuit between the parties involved!  

Still, people often ask, “What are the odds of picking a perfect bracket?”

This made me think of an old story. Years ago, at a start-up which eventually became one of the world’s largest technology firms, one of the founders was tasked with hiring a team of engineers and mathematicians. This founder was quick-minded and wanted bright people to help his company grow. Here is one of his favorite interview questions:

There are 64 teams playing in a single elimination tournament. If a team loses a game, they are eliminated from the tournament. Winners advance to the next round. How many games are needed to determine a winner?

The founder of the firm thought of this as a mental exercise. Most people would realize that 64 teams meant 32 games in the first round. Thus, during the pressure of an interview, applicants would have to think quickly:

– 32 games in round one, plus

– 16 games in round two, (equals 48), plus

– eight games in round three, (equals 56), plus

– four games in round four, (equals 60), plus

– two games in round five, (equals 62 — which is the semifinals — or final four), plus

– one game in round six, (for a total of 63 games), with this last game being the final to determine a champion

To rate his prospective associates, the founder took out his watch to time the applicant. He was surprised when one of the applicants blurted out, “63 games” before he could start timing the individual. That applicant, who became one of the founder’s best friends, explained his logic, “We’re determining a champion from 64 teams, so one team will go undefeated, and 63 teams will suffer a defeat. Thus, it is logical that 63 games are needed to determine the champion.” Math and logic can explain many things.

As we know, this is the format of the NCAA Men’s Basketball Tournament. Today, there are several additional twists such as a few teams needing wins to gain the final berths in the tournament.  However, many bracket pools continue with the 64-team bracket. To pick a perfect bracket, we know that we need to pick 63 games correctly. Without any knowledge of basketball, we can compute that the number of ways to pick the brackets is:

– two times two (equals four)

– times two (equals eight),

– times two (equals 16) and so on (a total of  63 times).

This is equivalent to “2 raised to the power 63″ — or “2 ^ 63.” This number grows large very quickly. As an aside, some people have said that they can win $1,000,000 by placing a $125 bet and winning 13 bets in a row while rolling the winnings into the next bet. In any case, two raised to the power 63 is a number much larger than a million. A very large number. It is so huge that it is easiest expressed in scientific notation, where we count the number of zeros:

2 ^ 63 = 9.2 x 10^18, or 9.2 E+18

Or roughly, this is a nine, followed by 18 zeroes! As a comparison, one million has six zeroes, and one billion has nine zeroes. Thus, this number, having 18 zeros — is more than a “billion billion!”

Are the odds really worse than a “billion billion” — or 9.2 E+18 — to one? This number assumes no knowledge of basketball or the NCAA Tournament and gives the number of different brackets people can pick. However, we know certain things, such as the fact that no 16th seed has ever beaten a no. 1 seed in the history of the tournament. Similarly, no. 2 seeds hold an overwhelming advantage over 15th seeds.

In addition, once the brackets are set, fans can estimate the chances of teams advancing in the tournament. Of course, there is still a world of randomness, so that the odds of picking a perfect bracket remain very high. Based on various assumptions, the odds of picking a perfect bracket has been estimated to be anywhere from 30-150 billion to one.

While the odds of picking a perfect bracket seem overwhelming, I will discuss methods to help you outdo your coworkers and friends in your office pool as the NCAA Men’s Tournament approaches. Let the madness begin!

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Carlton Chin is a fund manager, MIT-trained quantitative analyst, and co-author of “Who Will Win the Big Game?" He has been quoted by the Wall St. Journal, New York Times, and ESPN -- and worked with the Sacramento Kings on the 2014 NBA Draft.