NCAA Tournament PossibilitiesDate: 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/ |
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