7· In the NCAA basketball tournament, 64 teams compete for the championship (rea

Question

7· In the NCAA basketball tournament, 64 teams compete for the championship (really, it's 68, but we'll ignore the 4 "play-in" teams). Below is the bracket at the opening round for the 2018 men's tournament if you are unfamiliar with this tournament. It is popular for people to participate in competitions where brackets are filled out before the tournament. Prizes are often given after the tournament to those whose predictions most closely match what actually a. It is often claimed that the probability of filling out a perfect bracket is one out of 2.9 What assumptions are being made to come up with one out of 9.2 quintillion? happened. (3 points) quintillion (1/9,223,372,036,854,775,808). Show how this probability is calculated b. Many mathematicians and statisticians claim that this assumption is not reasonable and the probability of filling out a perfect bracket is actually larger than one out of 9.2 quintillion (i.e. it's more likely to happen). What are some possible factors that would C. invalidate the assumptions you listed in b? How would these factors increase the probability of filling out a perfect bracket?

Explanation / Answer

a. There are 64 teams and its a knockout tournament, so there will be total of 32+16+8+4+2+1 = 63 games to predict. Each game is played between 2 teams, so there are 2 choices for predicting the winner for each game so probablity of a team winning is 1/2. Therefore for 63 games we have 2*2*2*2.....*2 (63 times) which is equal to 2^63 (2 raised to power 63), which equals 9.2 quintillion (9,223,372,036,854,775,808) possibilities. A perfect bracket is when one predicts the correct winner for each of the 63 games. Which means there only 1 desired outcome(bracket containing winner of each of 63 games). Hence the probablity of filling a perfect score is 1 / 9,223,372,036,854,775,808

b. The assumption made to reach this 9.2 quintillion figure is that for each game both the teams have equally likely chance of winning. Meaning that both teams have 50-50% chance of winning. SO basically like a coin toss where probablity of getting a head, or a tail is 50%.

c. The above assumption means that both the teams have equal chance of winning a game. So according to assumption if a game is played between top seeded team and last seeded team, i.e games between 1st seeded team and 64th seeded team both teams have equal chance of winning. But in actual that is very unlikely as the 1st seeded team will almost always defeat the 64th seeded team, hence the 1st seeded team have a much higher chance of wining the game. Same will apply for every match, chance of winning by the team seeded higher will always be more than the chance of winning of lower seeded team in a game. These factors invalidate our assumption. Hence for each game probablity of predicting a winner will increase from assumed probablity 1/2. Since for each game probablity of predicting a winner is increased this will increase the total probablity of filling out the perfect bracket.

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