시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
3 초 128 MB 67 39 25 59.524%

문제

“영웅이여, 대 마상시합에 온 것을 환영합니다!” – 심판관 트루하트

쿠키런을 즐겨 플레이하는 샤바칸은 매일같이 하트가 충전되기를 기다리던 중, 한 TCG 장르의 게임을 접한다. 이 게임은 양 플레이어가 자신이 가진 카드 중 가장 좋은 30장을 순서대로 늘어놓아 자랑하며 경쟁하고, 여기서 번 골드를 이용해 새로운 카드를 구매하는 컨텐츠로 이루어져 있는데, 구입하는 새로운 카드는 무작위로 나타난다. 존재하는 카드의 등급은 다음과 같다.

  • 일반 카드
  • 희귀 카드
  • 영웅 카드
  • 전설 카드

열심히 골드를 모은 샤바칸은 N장의 카드를 구입하려고 한다. 샤바칸은 구입한 각 N장의 카드의 등급을 하나의 이진 문자열로 표현하고자 한다. 즉, 서로 다른 4^N가지의 등급 순열에 대해 각각 대응되는 이진 문자열이 존재해야 하며, 문자열의 끝을 명확히 알 수 있도록 이들 중 어떠한 문자열도 다른 문자열의 접두사(prefix)가 되어서는 안 된다.

구입한 카드에서 발생하는 등급의 확률분포가 주어질 때, 어떠한 N장의 등급 순열에서 이진 문자열로 변환하는 표현 방법에 대해, 우리는 샤바칸이 각 이진 문자열을 사용할 확률을 구할 수 있다. 이를 통해 우리는 샤바칸이 사용할 이진 문자열의 길이의 기댓값을 구할 수 있다.

구입한 카드에서 발생하는 등급의 확률분포가 주어질 때, 샤바칸이 사용할 이진 문자열의 길이의 기댓값의 최솟값을 구하시오. 

입력

입력의 첫 줄에는 구입하는 카드의 수 N (1 ≤ N ≤ 20)이 주어진다.

입력의 둘째 줄에는 구입한 카드에서 발생하는 일반 카드, 희귀 카드, 영웅 카드, 전설 카드의 확률 분포 PC, PR, PE, PL가 주어진다. 여기서 PC+PR+PE+PL=1임이 보장된다. 

출력

출력의 첫 줄에는 샤바칸이 사용할 이진 문자열의 길이의 기댓값의 최솟값을 출력한다.

절대 또는 상대 오차가 10-4이하인 경우 정답으로 인정된다

예제 입력 1

2
0.9 0.049999 0.05 0.000001

예제 출력 1

1.457510

예제 입력 2

20
0.25 0.25 0.25 0.25

예제 출력 2

40.000000
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Of course, this is no easy task. They have deployed many small devices around the world, designed to take periodic measurements of the local weather conditions. These are cheap devices with somewhat restricted capabilities. Every day they observe which of the four standard kinds of weather occurred: Sunny, Cloudy, Rainy, or Frogs. After every n of these observations have been made, the results are reported to the main server for analysis. However, the massive number of devices has caused the available communication bandwidth to be overloaded. The Association needs your help to come up with a method of compressing these reports into fewer bits.<\/p>\r\n\r\n<p>For a particular device&rsquo;s location, you may assume that the weather each day is an independent random event, and you are given the predicted probabilities of the four possible weather types. Each of the 4<sup>n<\/sup>&nbsp;possible weather reports for a device must be encoded as a unique sequence of bits, such that no sequence is a prefix of any other sequence (an important property, or else the server would not know when each sequence ends). The goal is to use an encoding that minimizes the expected number of transmitted bits.<\/p>\r\n","input":"<p>The first line of input contains an integer 1 &le; n &le; 20, the number of observations that go into each report. The second line contains four positive floating-point numbers, p<sub>sunny<\/sub>, p<sub>cloudy<\/sub>, p<sub>rainy<\/sub>, and pfrogs, representing the respective weather probabilities. These probabilities have at most 6 digits after the decimal point and sum to 1.<\/p>\r\n","output":"<p>Display the minimum expected number of bits in the encoding of a report, with an absolute or relative error of at most 10<sup>&minus;4<\/sup>.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]