Editorial for ICPC PACNW 2016 K - Tournament Wins
Submitting an official solution before solving the problem yourself is a bannable offence.
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 people, you win your first game iff you are the better player in your -player bracket, your second game iff you are the best player in your -player bracket, and your third game iff you are the best player among all people. If we can find the probability that you win your 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 game? Well, there are players in all, players in this particular bracket, and players better than you (or equivalently, players worse than you). You win in the event that of the other players in your bracket, all of them are chosen from the players that are worse than you. Since all brackets are equally likely, this happens with probability:
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:
- We can work with logarithms of factorials instead of factorials directly.
- We can do some simplification of the expression by cancelling common terms.
Comments
bruh isn't the formula obviously wrong? the numerator is larger than the denominator :thinking: The correct formula should be
fixed