# March Madness Probabilities

Odds are, you're not going to win Warren Buffet's billion dollar NCAA tournament bracket challenge, but you might win the office pool with some knowledge of the teams' rankings and a lot of luck. Just how much luck do you need? Guessing blindly, the probability of filling out a perfect bracket is

(1/2)^63 = 1 / 9,223,372,036,854,775,808

or about 1 out of 9.2 quintillion. This number comes the fact that there are 63 slots to fill out on the NCAA bracket and assumes that each game's outcome is a 50-50 toss-up. In real terms, every person on Earth would need to fill out 1,293,600,567 distinct March Madness brackets in order for there to be a winner.

In reality, the probability of each match is not 50-50. For example, in the first round when the highest seed teams (1) play the lowest seed teams (16), the highest seed teams have a nearly 100% chance of winning based on historical data. By analyzing historical data of how differently seeded teams fare against one another, you could derive a more accurate probability. In the YouTube video below, a mathematics professor from DePaul University shows that the probability of a basketball expert getting a perfect bracket is "only" 1 in 128 billion, rather than the 9.2 quintillion number often cited.

For fun, this article shows probabilities of getting a bracket partially correct assuming an even 50-50 chance for each game.

The video above shows Professor Jeffrey Bergen's derivation of his 1 in 128 billion figure for the likelihood of a men's college basketball expert getting a perfect March Madness bracket.

## Probability of Getting All the 2nd Round Games Correct

The second round has 32 games consisting of eight games played in four regions. In the NCAA tournament, the actual "first" round is a play-in tournament for four of the low-seed spots. Nobody really cares about those games because low-seed teams never make it far in the bracket.

In the second round, in each region, the 1-seed team plays the 16-seed team, 2 plays 15, 3 plays 14, 4 plays 13, 5 plays 12, 6 plays 11, 7 plays 10, and 8 plays 9. Assuming that each team has a 50-50 chance of winning, the probability of picking all 32 outcomes correctly is

(1/2)^32 = 1 / 4,294,967,296

So it's about 1 out of 4.3 billion. What if we assume that the 1s have a 100% chance of beating the 16s, as has been the case historically? This means four games are decided and the other 28 are still toss-ups. The probability of making a perfect selection is then

(1/2)^28 = 1/ 268,435,456

In other words, 1 out of 268 million. A little better than the odds of the first calculation, but your chances of winning the lottery are still better.

## Probability of Getting All the 3rd Round Games Correct

What about the probability of getting all the third round games correct, regardless of whether or not all your second round picks were right? Since the NCAA March Madness tournament is done by direct elimination, there are 16 games in the third round, half the number in the second round. For each spot in the third round, there are 64/16 = 4 teams vying for it. This means your probability of guessing the the third round winners correctly is 1/4 for each game. Thus, the probability of picking all 16 third round games correctly is

(1/4)^16 = 1/ 4,294,967,296

This is the same 1 in 4.3 billion figure computed in the previous section. Note that this is not computing the odds of getting both the second and first rounds correct. It is is quite possible to get correctly pick the winner of a third round game without correctly picking both winners of the previous two games from the second round.

## Probability of Getting All 2nd AND 3nd Round Games

Since the second and third rounds have a total of 32 + 16 = 48 games, the probability of correctly guessing all their outcomes, assuming a 50-50 chance for each team, is

(1/2)^48 = 1 / 281,474,976,710,656

That is 1 in 281 trillion. In comparison, the probability of winning the Powerball lottery jackpot *and* getting killed by a grizzly bear at Yellowstone National Park is only slight less:

(1/175223510)(1/1900000) = 1 / 332,924,669,000,000

Or 1 in 333 trillion. So you can pretty much forget about making a perfect bracket selection for the second and third rounds.

## Probability of Guessing the Final Four

Each team that makes it to the final four has to beat the 15 other teams in its region, which means each team in a region has a 1/16 chance of making it to the final four (under the perhaps faulty assumption that the games are all toss-ups.) So the probability of picking all four correctly is

(1/16)^4 = 1 / 65,536

In comparison, the odds of getting injured by a toilet in the US are roughly 1 in 6,265. This means you are 10 times more likely to need medical attention after visiting the restroom than you are to correctly pick the NCAA final four.

## Probablility of Picking the NCAA Champion

With 64 teams starting the bracket, and assuming they all have an equal chance of winning, the probability of guessing the champion is just 1/64, or 1.5625%. Of course, it's probably not valid to assume they all have the same chance of winning. If you discount the four 16-seed teams and the four 15-seed teams, then the pool of potential champions is 64 - 4 - 4 = 56. The likelihood of correctly picking the champion from these remaining teams is 1/56 ≈ 1.7857%.

## Comments

My husband is smart like you. And he loves bb. And he is in the office pool thing for ... something... I don't know. Anyway, He had me read this to him, but I have laryngitis and I couldn't finish it so I e mailed it to him. He wins a new set of golf clubs, I think, if he wins, so I will let you know how he fares. You're very smart. I like that. :)

May I cite your work for my research paper?

Thanks :)

UCONN was kind of a surprise...

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