AP Statistics Lectures

by Arnold Kling

March Madness

The NCAA basketball tournament consists of four brackets with 16 teams each. Within each bracket, one team wins (we say they make it to the Final Four). The Final Four play a two-round tournament to determine the champion. The goal of this exercise is to simulate the tournament using random numbers. That is, for each game, you pick a random two-digit number from a table or using the random number function of the calculator, and based on the number chosen, either the favored team wins or there is an upset. You should choose the odds for an upset. For example, if you say that the chance of an upset in a particular game is 30 percent, then if the random number is 1 through 30 you have an upset, and if it is 31 through 00 there is no upset.

In each bracket, the teams are seeded 1 through 16. Let me call the number 1 and 2 seeds the A teams, the 3 through 8 seeds the B teams, the 9 through 14 teams the C teams, and the 15th and 16th seeds the D teams. Since there are four brackets, there are 8 A teams, 24 B teams, 24 C teams, and 8 D teams. In the first round, the A teams play D teams, and the B teams play C teams. Each round, winners advance and losers are eliminated.

I found some data on past tournament performance at About.com, and based on that information, I would say the following:

- In the first round, D teams essentially never win. But usually 3 of the 24 C teams win.
- Usually, one of the 24 C teams wins two games, but a C team almost never wins four games.
- Almost always, at least one B team wins 4 games, to reach the final four. Almost always, at least two A teams win four games. A typical Final Four will consist of two A teams and two B teams.

Use that information to decide on a probability of an upset in an A vs. B game, an A vs. C game, and a B vs. C game. (We can set the probability of a D winning a game to zero.) Assume that when two roughly equivalent teams meet (A vs. A or B vs. B), the probability of an upset is 50 percent.

Simulate the tournament, including the two rounds of the final four. How many A, B, and C teams made it to your final four? Did an A team win the championship (over 80 percent of the time that happens)?

Try to simulate the tournament several times to get a feel for how the simulation works. Then think about what you can get out of simulation. Consider the following potential uses for simulation:

- Predicting which teams will achieve first-round upsets.
- Calculating the odds that a final four will consist of all teams that were seeded number 2 in their initial bracket.

Will simulation be an effective tool for doing (1), (2), both or neither? Write a paragraph justifying your answer.