What Are the Odds of Predicting a Perfect NCAA Tournament Bracket?
What are the odds of winning Warren Buffet's $1 billion prize for picking the perfect 2014 NCAA tournament bracket? Should Mr. Buffet be worried about his money? And how are the odds of winning related to the famous rice and chessboard problem? Keep on reading to find out!
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If you're like me, you don't have a billion dollars. And while neither you nor I need a billion dollars, we probably wouldn't turn it away either. Which is why I feel duty bound to inform you that right now we all have a chance to win $1 billion from Warren Buffet and his Berkshire Hathaway company.
And the best part is that winning Mr. Buffet's bet seems like it should be relatively easy—all you have to do is come up with the perfect bracket for this year's 64 team NCAA basketball tournament. Shouldn't be too hard, right?…Right?
Spoiler alert: Wrong!
But how hard is it? And how could this bet about a college basketball tournament possibly be related to a famous mathematical fable about a chessboard and some grains of rice? Those are exactly the questions we'll be answering today..
How Many Games in a Tournament?
Before we get to the odds behind Mr. Buffet's $1 billion offer, let's take a minute to talk about the answer to the puzzle posed at the end of last week's episode by math fan Raleigh, who writes:
"The manager at a country club had the job of scheduling a single-elimination tennis championship for 100 players. When asked, 'How many games will they play?' the manager dutifully started a list of pairs, available courts, winners, losers, etc., and got hopelessly bogged down. BUT, the answer is easy if you approach it properly."
And what is that proper approach that makes things so easy? As Raleigh writes:
"Every game has one loser. And every player loses exactly once except the champion. Thus, in a 100 player tournament, there must be 100 - 1 = 99 games to give us 99 losers and 1 winner."
As luck (or good planning) would have it, this idea comes in handy for our understanding of the odds behind perfect NCAA tournament brackets and billion dollar bets. And that's because in order to understand the odds, we need to know how many games are played amongst the 64 teams making up the final bracket. Using our new knowledge about such things, it's clear that 63 teams must lose in order to get 1 winner, which means that there must be 63 games in the NCAA tournament.
NCAA Tournament Bracket Odds
So what are the odds of picking a perfect bracket and winning that $1 billion? Let's say you know absolutely nothing about basketball and simply guess the winner of each game at random. In that case, you have a 50-50 shot of choosing the winner of any particular game. Since there are 2 possible outcomes of the first game and every game thereafter, and since you would have no insight about the probable winner of any game, there are 263 possible brackets you could come up with.
How many is that? It's an astoundingly gargantuan number. According to WolframAlpha, the number of grains of sand on Earth is between 10 and 100,000 times the number 263. In other words, there is one possible NCAA tournament bracket for perhaps every 10 grains of sand on the planet. Even if there's one possible tournament bracket for every 100,000 grains of sand, that's still an enormous number of brackets.
In the NCAA tournament, there are 263 possible brackets.
In case you're wondering, the actual value of the number 263 is about 9.2 x 1018—aka, 9.2 quintillion or 9.2 billion billion. So, yeah, it's a big number. Kind of boggles the mind to think there are that many possible tournament brackets, no?!
So what does this have to do with rice and a chessboard?