There are 9,223,372,036,854,775,808 (or 2**63) possible brackets. If each game is viewed as a toss-up,
then each of these brackets are equally likely to occur. However, basketball aficionados know that since 1985,
a No. 1 seed has lost to a No. 16 seed only twice (in 2018 and 2023), while the No. 8 - No. 9 game teams are evenly matched.
This means that not all games are toss-ups, and that using some basketball knowledge can improve the odds of filling in a perfect bracket.
So can be compute the odds of a perfect bracket iteratively by considering each round? The following table estimates these odds, assuming that each game is a toss-up, or each seed will perform exactly as it has historically.performed.
We can use this information to estimate the odds of picking a perfect bracket. If all games are toss-ups, then the odds are 9,223,372,036,854,775,807 (or (2**63)-1) to 1. If all seeds are expected to perform exactly as they have historically and taking into account the four regions acting independently, then using the third column in the table, the most optimistic estimated odds of getting all the teams correct in their respective regions is just over 8 billion to 1, a true perfect bracket. Here are the most optimistic odds broken down by rounds.
| Round | Toss-up | Historical Performance, Teams | Round of 64 (32 games correct) | 4,294,967,295 | 26.8 thousand |
|---|---|---|
| Round of 32 (16 games correct, given that the Round of 64 is correct) | 65,535 | 492 |
| Sweet Sixteen (8 games correct, given that the Round of 32 is correct) | 255 | 9 |
| Elite 8 (4 games correct, given that the Sweet Sixteen is correct) | 15 | 7 |
| Final Four (2 games correct, given that the Elite Eight is correct) | 3 | 3 |
| National Champion (1 game correct, given that the National Finalists are correct) | 1 | 1 |
| All Rounds (63 games correct) | 9,223,372,036,854,775,807 | 8.3 billion |
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