8. Challenge Problem March Madness. March Madness is a single elimination tourna

Question

8. Challenge Problem March Madness. March Madness is a single elimination tournament with 64 teams. Determine the number of distinct brackets (show work). Hint: Figure out how many ways there are to choose winners at each round, and use the Fundamental Theorem of Counting to obtain your final answer. Now assume that you have a computer program which generates distinct brackets at random a million times every second for 2 weeks. What is the probability that your computer program generates a perfect bracket?

Explanation / Answer

The NCAA’s March Madness is a basketball single-elimination tournament in which 64 teams play.

In the first round, there are 64 teams and therefore 32 games, each with 2 possible outcomes. This provides for a total of 232 possibilities.

In the second round, there are 32 teams left so now there are 16 games, again each with 2 possible outcomes, for a total of 216 possibilities.

In the 3rd round or regional semifinals, there will be only 16 teams left and therefore 8 games possible, each game with 2 possible outcomes, at this round there are 28 possibilities.

In the 4th round or regional finals, there will be only 8 teams left and therefore 4games possible each game with 2 possible outcomes, again at this round there are 24 possibilities.

In the 5th round or National Semifinals, there are 4 teams left so only 2 games possible each game having 2 possible outcomes, at this round there are 22 possibilities.

In the 6th round or National Final, there are 2 teams playing in the final, so possibility of either team winning so at this final round there are 21 =2 possibilities.

Now using fundamental theorem of counting, these possibilities at each round will be multiplied to get overall possibilities or Brackets. So total number of distinct brackets are (232)x(216)x(28)x(24)x(22)x(21)= 263=9,223,372,036,854,775,808 possible brackets.

Now coming to the second question, Since a computer program is generating distinct brackets at random a million times per second for 2 weeks so there will be 106x3600=36x108 distinct brackets generated every hour, therefore (36x108)x24=864x108 random brackets generated. This leads to (864x108)x14=12096x108 distinct brackets generated in 2 weeks randomly.

The question can be regarded as selecting a specified unit of a population containing N distinct units in a random sample of n units using without replacement scheme(since every time distinct bracket is generated). This probability is given by n/N. so in this case, N=9,223,372,036,854,775,808 and n=1,209,600,000,000. Upon solving we get approximately 0.000000131145 or 1.31145x10-7.

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