# Probabilities for Winning the NCAA Tournament: Sweet Sixteen Edition

Now that 52 games of the NCAA Tournament are in the books, we come to the Sweet 16. Despite all the crazy upsets and close calls, we are left with 15 power conference teams and one perennial power-house of a “mid-major” team in Gonzaga. Using the resulting 52 games, we are able to update the probabilities for each team winning the NCAA Tournament. Using the original probabilities, we are able to update the conditionals.

The method is relatively simple. Before the tournament started, the probability that Kansas made it to the Elite Eight was written via the law of total probability as:

P(Kansas beats Maryland)P(Kansas Makes Sweet 16)P(Maryland makes Sweet 16) +

P(Kansas beats California)P(Kansas Makes Sweet 16)P(California makes Sweet 16) +

P(Kansas beats Hawaii)P(Kansas Makes Sweet 16)P(Hawaii makes Sweet 16) +

P(Kansas beats South Dakota State)P(Kansas Makes Sweet 16)P(South Dakota State makes Sweet 16).

The updated conditional distribution, given that Kansas and Maryland made the Sweet 16 is:

P(Kansas Makes Elite 8 | Kansas and Maryland are in the Sweet 16) = P( Kansas beats Maryland) P(Kansas Makes Sweet 16)P(Maryland Makes Sweet 16) / P(Kansas Makes Sweet 16)P(Maryland Makes Sweet 16)

Completing these updates and re-running the Markov Chain Monte Carlo, we are able to update the probabilities for each team winning the NCAA Tournament:

## Probability of Winning the NCAA Tournament

 Team Previous Updated Virginia 7.4450 13.77508 North Carolina 7.2837 12.39097 Kansas 8.5790 10.82387 Villanova 6.3156 8.29242 Oregon 6.6224 8.29226 Oklahoma 5.9477 7.73610 Iowa State 2.6482 6.74002 Miami 3.9689 5.72485 Texas A&M 3.8008 5.41548 Indiana 2.0066 5.19359 Duke 2.7659 4.49773 Notre Dame 1.0331 4.49109 Maryland 1.8846 3.39633 Wisconsin 0.6325 1.16051 Gonzaga 0.0815 1.06463 Syracuse 0.1426 1.00507

The simulation now shows that despite Kansas winning their respective games, the Virginia Cavaliers have the best chance of winning the tournament. This is partly due to the fact that their opponent, Iowa State, is a road-block to a low-probability opponent in either Gonzaga or Syracuse. Kansas’ probability ever so slightly changed due to the fact that the entire region is effectively a Top-4 (with #5 Maryland being the “outsider”). Meanwhile, North Carolina’s and Virginia’s regions have been busted up thanks to Gonzaga and Stephen F. Austin University.

## Highest Probability Outcomes Updated

Due to the changes, the current highest probability finish to the tournament is as follows:

Elite Eight: Kansas vs. Villanova; North Carolina vs. Notre Dame; Virginia vs. Gonzaga; Oregon vs. Oklahoma

Final Four: Kansas vs. Oregon; Virginia vs. North Carolina

Championship: Virginia vs. Kansas

Champion: Virginia

While this is the highest probability path, I’m still holding out for a personal belief that North Carolina will upend Kansas in the championship. I could be wrong, though. My bracket did have Purdue and West Virginia in the Elite Eight and we see where that has gotten me (near last in the office pool). How do you think the tournament will end?

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