Sample Space for Conventional Knock-Out Tournament

[Alexsandro]

Somebody could help me with this question?

A conventional knock-out tournament begins with 2^n competitors and has n rounds. There are no play-offs for the positions 2, 3, ..., 2^(n)-1, and the initial table of draws is specified. Give a concise description of the sample space of all possible outcomes.

thanks,

Alexsandro

 

[HallsofIvy]

That depends upon what you consider an outcome! If you are only asking "who won the tournament", then the set of possible outcomes is precisely the set of players- because anyone of them could win the tournament. If you are thinking of the "rankings" awarded by this tournament: the winner, the person who only lost in the final round (and so comes in second), the two people who lost to those first two (and so share third and fourth places), etc. that's a much more complicated question!

 

[tacman]

The sample space for a conventional knock-out tournament with 2^n competitors and n rounds can be described as a set of all possible outcomes or paths that the tournament can take. Each outcome represents a unique combination of winners and losers in each round, ultimately leading to a single champion. The sample space would include all possible combinations of winners and losers in each round, with the total number of outcomes being 2^n, as there are 2 possible outcomes (win or lose) for each competitor in each round. The initial table of draws would determine the initial matchups and would be a key factor in determining the sample space. Ultimately, the sample space would represent all possible paths that the tournament could take, starting from the initial round and ending with the final champion.