NCAA Tournament Possibilities

Date: 03/14/2001 at 15:28:45
From: Jill Paskoff
Subject: NCAA tournament possibilities

If I had an NCAA Tournament office pool, where each person fills out 
the brackets by selecting who he or she thinks will win each game, 
how many possible combinations of choices are there?

Date: 03/14/2001 at 17:12:28
From: Doctor Paul
Subject: Re: NCAA tournament possibilities

A great question at a wonderful time of the year!

First notice that the 64 teams play 63 total games: 32 games in the 
first round, 16 in the second round, 8 in the 3rd round, 4 in the 
regional finals, 2 in the final four, and then the national 
championship game.

32+16+8+4+2+1= 63

Now let's answer an easier question.

If there were four teams, and they played three games, how many 
different ways would there be to fill out a bracket?  You can write 
them down. There are only eight of them. Where do that eight come 
from?  Well, there are three games, and you have two possible choices 
for each game. Hence, 2^3 = 8 possibilities.

Now back to the real tournament. Since there are 63 games to be 
played, and you have two choices at each stage in your bracket, there 
are 2^63 different ways to fill out the bracket.

2^63 = 9,223,372,036,854,775,808

That's more than nine quintillion possibilities.

Perhaps now you can see why CNNSI is so willing to give away ten 
million dollars to the person that fills out a perfect bracket. It's 
just not likely to happen.

Even if every person in America filled one out (300 million people), 
the probability of someone winning is:

300,000,000
----------- = .00000000003253
  2^63

I don't think anyone is going to fill out a perfect bracket any time 
soon.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/   
    

Date: 03/14/2006 at 12:24:36
From: Doctor Ian
Subject: NCAA tournament possibilities

A lot of people have written to us, asking whether this really takes into 
account the structure of the tournament, the fact that once a team is 
eliminated it can't play again, and so on.  A thought experiment can help 
show why 2^63 is the correct answer.  

Imagine that you're going to fill in a bracket by flipping a coin.  Each 
time you get heads, you fill in the first team (the one above, or to the 
left).  If you get tails, you fill in the second team. 

Each possible sequence of 63 coin flips corresponds to a unique way of
filling in a bracket.  So the number of ways to fill in a bracket has
to be the same as the number of unique sequences of 63 coin flips, which
is 2^63.

I hope this helps clear up any confusion. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/