# The odds of a perfect bracket are roughly a billion to 1

This time of year it is widely quoted that odds of picking a prefect bracket are 9,2 quintillion to one. In scientific notation that is 9.2 x 1018 or if you like writing out all the digits it is 9,223,372,036,854,775,808 to 1. That number is 263, i.e., the chance that you succeed if you flip a coin to make every pick.

If you know a little then you can do much better than this, by say taking into account the fact that a 16 seed has never beaten a one-seed. In a story widely quoted last year “Duke math professor Jonathan Mattingly calculated the odds of picking all 32 games correctly is actually one in 2.4 trillion.” He doesn’t give any details, but I don’t know why I should trust a person who doesn’t know there are 63 games in the tournament.

Using a different approach, DePaul mathematician Jay Bergen  calculated the odds at one in 128 billion. His youtube video from four years ago https://www.youtube.com/watch?v=O6Smkv11Mj4 is entertaining but light on details.

Here I will argue that the odds are closer to one billion to 1. The key to my calculation of the probability of a perfect bracket is use data from outcomes of the first round games for 20 years of NCAA 64 team tournaments. The columns give the match up, the number of times the two teams won and the percentage

1-16                 80-0                 1

2-15                 76-4                 0.95

3-14                 67-13               0.8375

4-13                 64-16               0.8

5-12                 54-26               0.675

6-11                 56-24               0.7

7-10                 48-32               0.6

8-9                   37-43               0.5375

From this we see that if we pick the 9 seed to “upset” the #8 but in all other case pick the higher seed then we will pick all 8 games correctly with probability 0.09699 or about 0.1, compared to the 1/256 chance you would have by guessing.

Not having data for the other seven games, I will make the rash but simple assumption that picking these seven games is also 0.1. Combining our two estimates, we see that the probability of perfectly predicting a regional tournament is 0.01. All four regional tournaments can then be done with probability 10-8. There are three games to pick the champion from the final four. If we simply guess at this point we have a 1 in 8 chance ad a final answer of about 1 in a billion.

To argue that this number is reasonable, lets take a look at what happened in the 2015 bracket challenge. 320 points are up for grabs in each round: 10 points for each 32 first round games (the play in or “first four games” are ignored), 20 for each of the 16 second round games, and so on until picking the champion gives you 320 points. The top ranked bracket had

27 x 10 + 14 x 20 + 8 x 40 + 4 x 80 + 2 x 160 + 1 x 320 = 1830 points out of 1920.

This person missed 5 first round and 2 second round games. There are a number of other people with scores of 1800 or more, so it is not too far fetched to believe if the number of entries was increased by 27 = 128 we might have a perfect bracket. The last calculation is a little dubious but if the true odds were 4.6 trillion to one or event 128 billion to 1, it is doubtful one of 11 million entrants would get this close.

With some more work one could collect data on how often an ith seed beats a jth seed when they meet in a regional tournament or perhaps you could convince ESPN to see how many of its 11 million entrants managed to pick a regional tournament correctly. But that is too much work for a lazy person like myself on a beautiful day during Spring Break.