Editorial for ICPC PACNW 2016 K - Tournament Wins


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You win the final game of a tournament bracket iff you are the highest ranked player in that bracket. For example, in a tournament with 8 people, you win your first game iff you are the better player in your 2-player bracket, your second game iff you are the best player in your 4-player bracket, and your third game iff you are the best player among all 8 people. If we can find the probability that you win your i^\text{th} game using this, then we can use the linearity of expectation to sum these probabilities up to get our final answer.

So what is the probability that you win your i^\text{th} game? Well, there are 2^k players in all, 2^i players in this particular bracket, and r-1 players better than you (or equivalently, 2^k-r players worse than you). You win in the event that of the 2^i-1 other players in your bracket, all of them are chosen from the 2^k-r players that are worse than you. Since all brackets are equally likely, this happens with probability:

\displaystyle \binom{2^k-r}{2^i-1} / \binom{2^k-1}{2^i-1}

It's dangerous to compute these two values directly and then divide them, since these are both very big numbers. We have two options for dealing with this:

  1. We can work with logarithms of factorials instead of factorials directly.
  2. We can do some simplification of the expression by cancelling common terms.

Comments


  • 0
    thomas_li  commented on July 27, 2024, 8:32 p.m.

    bruh isn't the formula obviously wrong? the numerator is larger than the denominator :thinking: The correct formula should be {2^k-r \choose 2^i-1} / {2^k-1 \choose 2^i-1}


    • 0
      d  commented on July 27, 2024, 9:34 p.m.

      fixed