Expert: Chanda Walker - 3/19/2009

Question
March is the month when the National Collegiate Basketball champion is determined.  Currently there are sixty-five teams in the men's tournament.  The 64th and 65th seeds play in an opening round.  The winner advances and the real tournament begins. The field of 64 teams is divided into four geographical regions. Each region has teams that are assigned a seed number of 1 through 16, with the best team in the region awarded the 1 seed.



Since the expansion to 64 teams in 1985, there has never been a case where all four number one seeds made it to the Final Four.  The closest it has ever come to this was in 1993 when three number one seeds (Kentucky, Michigan, North Carolina) and a number two seed (Kansas) made it to the Final Four.


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A regional is a single elimination tournament of 16 teams. Each round of the regional pairs two teams to play a game and the losing team is eliminated.  The champion of each regional goes to the Final Four.  Assume that every team is equally likely to win any game.  What is the probability that each of the number one seeds goes to the Final Four?  Express your answer as a common fraction.


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There are 30 states and the District of Columbia  represented in the men’s tournament.  Texas has 5 schools in the tournament and has the greatest representation of any state.  Each of the letters in TEXAS is placed in a bowl.  One letter at a time is selected at random without replacement.  What is the probability the letters will be selected in the order that correctly spells TEXAS? Express your answer as a common fraction.


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There are 65 schools entered in the men’s tournament and 64 schools entered in the women’s tournament.  Twenty-two of these schools have both a men’s team and a women’s team in their respective tournaments.  How many different schools are represented in the two tournaments

O.K.  Here's what I have so far.

1. 8 games   chances of winning each game 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/256 for each regional to advance to final four.  Is it 1/256 x 1/256 x 1/256 x 1/256 ?  I'm confused.

2. TEXAS 1/5 x 1/4 x 1/3 x 1/2 = 1/120

3. 65 - 22 = 43
  64 - 22 = 42
  43 + 22 + 42 = 107 schools

Answer
March is the month when the National Collegiate Basketball champion is determined.  Currently there are sixty-five teams in the men's tournament.  The 64th and 65th seeds play in an opening round.  The winner advances and the real tournament begins. The field of 64 teams is divided into four geographical regions. Each region has teams that are assigned a seed number of 1 through 16, with the best team in the region awarded the 1 seed.

Since the expansion to 64 teams in 1985, there has never been a case where all four number one seeds made it to the Final Four.  The closest it has ever come to this was in 1993 when three number one seeds (Kentucky, Michigan, North Carolina) and a number two seed (Kansas) made it to the Final Four.



1. 8 games   chances of winning each game 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/256 for each regional to advance to final four.  Is it 1/256 x 1/256 x 1/256 x 1/256 ?  I'm confused.


I don't think that is right.  There are only four rounds of play off's before we make it to the final four.  so it is only (1/2)^4 = 1/16.

Let me explain:

Round 1:  (1 2)    (3 4)    (5 6)   (7 8)   (9 10)  (11 12)  (13 14) (15 16)

Round 2:  (1 3) (5 7) (9 11) (13 15)

Round 3:  (1 5) (9 13)

Round 4: (1 9)

And for each of them to make it to the final four it is (1/16)^4

So the total probability is 1/65536.

However, in real life, the probabilities are much higher than that because the chances of the best team winning should be higher than 50%.

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There are 30 states and the District of Columbia  represented in the men’s tournament.  Texas has 5 schools in the tournament and has the greatest representation of any state.  Each of the letters in TEXAS is placed in a bowl.  One letter at a time is selected at random without replacement.  What is the probability the letters will be selected in the order that correctly spells TEXAS? Express your answer as a common fraction.

What a weird question!
Your answer is perfect though.

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There are 65 schools entered in the men’s tournament and 64 schools entered in the women’s tournament.  Twenty-two of these schools have both a men’s team and a women’s team in their respective tournaments.  How many different schools are represented in the two tournaments
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Perfect!  There are 107 schools.