Question

In January 2014, the company Quicken Loans made news by announcing that it would pay $1 billion to the person who submitted the perfect NCAA bracket for the men's Division I tournament that year. The company bought an insurance policy from Warren Buffett's holding company Berkshire Hathaway to cover the cost of any prize money.

Assume that everyone in the United States filled out many brackets and that all of the brackets produced in this way were different. How many brackets would each person have to fill out to make sure that altogether all of the possible outcomes of the 63 games where represented?

Use the fact that the population of the United States at this time was 317,622,403 people. Also, your answer should be a whole number since we can not make a portion of a bracket.

Answer 1

Possibile out of each game = 2

Total possible outcomes for 63 games = 2^63= 9,223,372,036,854,780,000 which is the total number of different ways to fill out a bracket if we ignore 4 play-in games.

Assumption: Expert's chances of having a perfect bracket can be approximated by likelihood that the higher seeds always win.Hence we need to calculate the chances for which top seeds always win.

1st round: based on historic data from http://mcubed.net/ncaab/seeds.shtml probability for the pairings of 1 vs 16 = 1 ( never lost) and 8 vs 9 = 0.51 ( always toss up)

Hence for each of the pairings, probability of higher seed winning is derived with a subtraction of 7% i.e. 0.07 (i.e. (1-0.51)/(8-1))

1 vs 16 = 1.00, 2 vs 15 = 0.93, 3 vs 14 = 0.86, 4 vs 13 = 0.79, 5 vs 12 = 0.72, 6 vs 11 = 0.65, 7 vs 10 = 0.58, 8 vs 9 = 0.51.

The probability of all eight higher seeds winning in given region in first round is their product = 0.087. As there are four regions, probability of winning all 32 first round games = (0.087)^4

2nd round: if all higher seeds win pairing would be 1 vs 8, 2 vs 7, 3 vs 6, 4 vs 5

For each game - average probability of higher seed winning is 0.65 from historic tournament data available at the above mentioned link

Hence, probability of correctly predicting 16 second round pairings = (0.087)^4 * (0.65)^16

3rd round: 1 vs 4 and 2 vs 4

Probability = 0.6 for each and for 8 third round pairings probability = (0.6)^8

After this, all higher seeds have one previous games and all left is 7 games involving 1,2 seeds. Fair to consider remaining 7 games as toss up and probability of correctly predicting = 0.5, probability for 7 games = (0.5)^7

For total of first 56 games probability = (0.087)^4*(0.65)^16*(0.60)^8*(0.5)^7 = approximately 1/131,011,448,943

Hence expert has approximately 1 in 131 billion chance of getting a perfect bracket

As population of United states = 317,622,403

Hence we could each fill out 412 of these likely brackets and split $1 billion 317 million ways