# 2019 NCAA Div. 1 Men’s Basketball Tournament (Probability, Statistics, Weighted Scoring, Powers of Two, Addition and Multiplication)

There are 347 colleges and universities which have division one basketball teams. The top 68 teams are voted by the ten-member selection committee made up of athletic directors and conference commissioners throughout Division I. The First Four round eliminates four teams then there is a bracket of 64 teams. Of the 64 teams, 32 are automatically selected if they win their conference championship. Then the first round of 64 teams will eliminate 32 teams; the second round of 32 teams will eliminate 16 teams; the third round called the Sweet 16 will eliminate 8 teams; the fourth round called the Elite 8 will eliminate 4 teams until the Final Four round will eliminate two teams until the Championship Game.

First the children noticed that each of these rounds is a power of two: 64 = 2^6; 32 = 2^5; 16 = 2^4; 8 = 2^3; 4 = 2^2; 2 = 2^1; and 1 = 2^0. We talked about the rankings by the selection committee and how a higher ranked team will have the lowest number rank (for example, the number 2 team in a region against the number 15 team in that region is a mismatch—the number 2 team will most likely win in 94% of the games). To practice scoring using my method, I had them look at the 2018 tournament where all of the teams were selected based on their ranking only (page 4 of the pdf). I circled each team who won their game and put an X if the team lost. In the First Four games, one point is awarded for each correct pick. In Round One—two points for each correct pick; Round Two—5 points; Round Three—10 points; Round Four—25 points; Round Five—50 points; and the final Round Six—100 points. The total points earned in 2018 by the selection committee rankings was 218 of a maximum of 528 points.

The children will follow the winning teams each week with me or on their own if they like and add up their total points after April 8th. The top three winners in each class will earn more Mathlete Dollars.

When the children made their choices we discussed how mathematicians make decisions based on probabilities and data. They could use the ranking system as well as the odds published by Las Vegas for legal betting (page 5 of the pdf). The final page of the pdf shows the percentage chance of each ranked team to win their game in each round. Since mathematical models are only predictive, they are not determinative meaning that there is no guarantee that the predictions will determine the outcome. Sometimes you just have to go with your intuition or maybe your parent’s alma mater.

Another opportunity to earn Mathlete Dollars is to calculate the payout on a \$100 bet on each team to win the Championship. For example, Gonzaga has 5:1 odds which means a \$100 bet will pay off \$500 if they win the tournament. If you chose Yale to win the championship at 1000:1 odds, a \$100 bet will pay off \$100,000. While this payout is tempting, it is almost guaranteed to lose your \$100. The smart money is on the first eight teams. Of course, only adults may place bets but children may wager a candy bar if they choose.

Finally, mathematicians always have been waiting for that basketball fan to choose all 63 winners correctly. I will pay \$1 million Mathlete Dollars to any of my students who can accomplish this task. Assuming no advantage for top seeded teams (every game, each team has an equal chance of winning), odds of picking a perfect NCAA tournament bracket are a staggering 1 in 2^63 or 9,223,372,036,854,775,808 (that’s 9.2 quintillion).

Adjusting probability based on seeding, a Duke math professor puts the odds of picking all 32 games correctly is actually one in 2.4 trillion.

Using a different formula, DePaul mathematician Jay Bergen calculated the odds at one in 128 billion. Either way, you have a better chance of becoming President of the United States than anyone on Earth has at filling out a perfect bracket.

Have fun.

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