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문제

20세기 중반에 사회학자들은 맥주에 관련된 학생들의 행동에 대한 연구를 시작했다. 연구의 가장 큰 성과는 학생들의 수업이 끝나고 난 이후에 호프집을 선택하는 복잡한 과정을 매우 정확하게 묘사하는 방법을 개발한 것이다.

학생들은 호프집을 투표를 통해서 결정한다. 사회학자들은 일부의 주도하는 학생들이 다른 학생들에게 눈치를 주는 점에 주목했다. 

호프집을 선택하는 과정은 다음과 같이 나타낼 수 있다.

  • 주도하는 학생들은 각자 선호하는 호프집을 가지고 있다. 이 학생들은 항상 그 호프집에 표를 던진다. 주도적인 학생 여러 명이 같은 호프집에 표를 던질 수도 있다.
  • 그 다음, 남은 학생들은 모두 주도적인 학생이 아니다. 이 학생들은 주도적인 학생들의 눈치를 보면서 확률적으로 투표를 하며. 그들이 어떤 특정 호프집에 투표를 할 확률은, 그 호프집이 그 때까지 받은 표의 수를 그 때 까지 투표한 사람의 수로 나눈 값이다.
  • 모든 사람이 투표를 하게 되면, 가장 높은 표를 얻은 호프집이 선택된다. 만약, 가장 높은 표를 받은 호프집이 여러 개인 경우에는, 임의로 선택한다. (이때, 각 호프집을 선택할 확률은 모두 같다)

예를 들어, 학생이 일곱 명, 호프집이 세 개가 있고, 그 중 다섯 명이 주도적인 학생인 경우를 생각해보자. 세 학생이 1번 호프집에 표를 던지고, 나머지 학생이 각각 2번과 3번 호프집에 표를 던졌다고 하자. 이제 각 호프집이 받은 표의 수는 (3, 1, 1)이 된다.

이제 주도적이지 않은 학생 두 명이 투표를 해야 한다. 각 학생은 차례대로 투표를 한다. 먼저, 첫 번째 학생이 1번 호프집을 선택할 확률은 3/5, 2번과 3번을 선택할 확률은 1/5이다. 이 학생이 3번 호프집을 선택하면, 각 호프집이 받은 표의 수는 (3, 1, 2)가 된다.

두 번째 학생이 1번 호프집을 선택할 확률은 1/2, 2번 호프집은 1/6, 3번 호프집은 1/3이 된다. 이 학생은 3번 호프집을 선택했다고 하면, 각 호프집이 받은 표의 수는 (3, 1, 3)이 된다. 1번 호프집과 3번 호프집의 표가 같기 때문에, 두 호프집은 1/2의 확률로 선택을 받게 된다.

학생의 수와, 호프집의 수, 그리고 모든 주도적인 학생들이 선택한 호프집이 주어진다. 이때, 각 호프집을 갈 확률을 구하는 프로그램을 작성하시오.

입력

입력은 여러 개의 테스트 케이스로 이루어져 있다.

각 테스트 케이스의 첫째 줄에는 맥주집의 수 n (n ≤ 5)과 학생의 수 k가 주어진다. (k ≤ 50) 둘째 줄에는 주도적인 학생의 투표가 끝난 후의 결과 a1, a2, ..., an이 주어진다. 항상 \( k \ge \sum_{i=1}^{n}{a_i}\)를 만족한다.

출력

각 테스트 케이스에 대해서, 각 맥주집이 선택받을 확률을 출력한다. 'pub i: percentage %'와 같이 출력하며, i는 맥주집의 번호, percentage는 그 맥주집을 선택할 확률을 반올림해서 소수점 둘째 자리까지 나타낸 값이다.

예제 입력 1

3 7
3 1 1

예제 출력 1

pub 1: 93.33 %
pub 2: 3.33 %
pub 3: 3.33 %
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(\uc774\ub54c, \uac01 \ud638\ud504\uc9d1\uc744 \uc120\ud0dd\ud560 \ud655\ub960\uc740 \ubaa8\ub450 \uac19\ub2e4)<\/li>\r\n<\/ul>\r\n\r\n<p>\uc608\ub97c \ub4e4\uc5b4, \ud559\uc0dd\uc774 \uc77c\uacf1 \uba85, \ud638\ud504\uc9d1\uc774 \uc138 \uac1c\uac00 \uc788\uace0, \uadf8 \uc911 \ub2e4\uc12f \uba85\uc774 \uc8fc\ub3c4\uc801\uc778 \ud559\uc0dd\uc778 \uacbd\uc6b0\ub97c \uc0dd\uac01\ud574\ubcf4\uc790. \uc138 \ud559\uc0dd\uc774 1\ubc88 \ud638\ud504\uc9d1\uc5d0 \ud45c\ub97c \ub358\uc9c0\uace0, \ub098\uba38\uc9c0 \ud559\uc0dd\uc774 \uac01\uac01 2\ubc88\uacfc 3\ubc88 \ud638\ud504\uc9d1\uc5d0 \ud45c\ub97c \ub358\uc84c\ub2e4\uace0 \ud558\uc790. \uc774\uc81c \uac01 \ud638\ud504\uc9d1\uc774 \ubc1b\uc740 \ud45c\uc758 \uc218\ub294 (3, 1, 1)\uc774 \ub41c\ub2e4.<\/p>\r\n\r\n<p>\uc774\uc81c \uc8fc\ub3c4\uc801\uc774\uc9c0 \uc54a\uc740 \ud559\uc0dd \ub450 \uba85\uc774 \ud22c\ud45c\ub97c \ud574\uc57c \ud55c\ub2e4. \uac01 \ud559\uc0dd\uc740 \ucc28\ub840\ub300\ub85c \ud22c\ud45c\ub97c \ud55c\ub2e4. \uba3c\uc800, \uccab \ubc88\uc9f8 \ud559\uc0dd\uc774 1\ubc88 \ud638\ud504\uc9d1\uc744 \uc120\ud0dd\ud560 \ud655\ub960\uc740 3\/5, 2\ubc88\uacfc 3\ubc88\uc744 \uc120\ud0dd\ud560 \ud655\ub960\uc740 1\/5\uc774\ub2e4. \uc774 \ud559\uc0dd\uc774 3\ubc88 \ud638\ud504\uc9d1\uc744 \uc120\ud0dd\ud558\uba74, \uac01 \ud638\ud504\uc9d1\uc774 \ubc1b\uc740 \ud45c\uc758 \uc218\ub294 (3, 1, 2)\uac00 \ub41c\ub2e4.<\/p>\r\n\r\n<p>\ub450 \ubc88\uc9f8 \ud559\uc0dd\uc774 1\ubc88 \ud638\ud504\uc9d1\uc744 \uc120\ud0dd\ud560 \ud655\ub960\uc740 1\/2, 2\ubc88 \ud638\ud504\uc9d1\uc740 1\/6, 3\ubc88 \ud638\ud504\uc9d1\uc740 1\/3\uc774 \ub41c\ub2e4. \uc774 \ud559\uc0dd\uc740 3\ubc88 \ud638\ud504\uc9d1\uc744 \uc120\ud0dd\ud588\ub2e4\uace0 \ud558\uba74, \uac01 \ud638\ud504\uc9d1\uc774 \ubc1b\uc740 \ud45c\uc758 \uc218\ub294 (3, 1, 3)\uc774 \ub41c\ub2e4. 1\ubc88 \ud638\ud504\uc9d1\uacfc 3\ubc88 \ud638\ud504\uc9d1\uc758 \ud45c\uac00 \uac19\uae30 \ub54c\ubb38\uc5d0, \ub450 \ud638\ud504\uc9d1\uc740 1\/2\uc758 \ud655\ub960\ub85c \uc120\ud0dd\uc744 \ubc1b\uac8c \ub41c\ub2e4.<\/p>\r\n\r\n<p>\ud559\uc0dd\uc758 \uc218\uc640, \ud638\ud504\uc9d1\uc758 \uc218, \uadf8\ub9ac\uace0 \ubaa8\ub4e0 \uc8fc\ub3c4\uc801\uc778 \ud559\uc0dd\ub4e4\uc774 \uc120\ud0dd\ud55c \ud638\ud504\uc9d1\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. \uc774\ub54c, \uac01 \ud638\ud504\uc9d1\uc744 \uac08 \ud655\ub960\uc744 \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n","input":"<p>\uc785\ub825\uc740 \uc5ec\ub7ec \uac1c\uc758 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4.<\/p>\r\n\r\n<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uccab\uc9f8 \uc904\uc5d0\ub294 \ub9e5\uc8fc\uc9d1\uc758 \uc218 n (n &le; 5)\uacfc \ud559\uc0dd\uc758 \uc218 k\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. (k &le; 50) \ub458\uc9f8 \uc904\uc5d0\ub294 \uc8fc\ub3c4\uc801\uc778 \ud559\uc0dd\uc758 \ud22c\ud45c\uac00 \ub05d\ub09c \ud6c4\uc758 \uacb0\uacfc a<sub>1<\/sub>, a<sub>2<\/sub>, ..., a<sub>n<\/sub>\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. \ud56d\uc0c1 \\( k \\ge \\sum_{i=1}^{n}{a_i}\\)\ub97c \ub9cc\uc871\ud55c\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc5d0 \ub300\ud574\uc11c, \uac01 \ub9e5\uc8fc\uc9d1\uc774 \uc120\ud0dd\ubc1b\uc744 \ud655\ub960\uc744 \ucd9c\ub825\ud55c\ub2e4. &#39;pub i: percentage %&#39;\uc640 \uac19\uc774 \ucd9c\ub825\ud558\uba70, i\ub294 \ub9e5\uc8fc\uc9d1\uc758 \ubc88\ud638, percentage\ub294 \uadf8 \ub9e5\uc8fc\uc9d1\uc744 \uc120\ud0dd\ud560 \ud655\ub960\uc744 \ubc18\uc62c\ub9bc\ud574\uc11c \uc18c\uc218\uc810 \ub458\uc9f8 \uc790\ub9ac\uae4c\uc9c0 \ub098\ud0c0\ub0b8 \uac12\uc774\ub2e4.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"3641","problem_lang":"1","title":"Beer Pressure","description":"<p>Ever since the middle of the 20th century, sociologists have been studying the fascinating topic of beer-related behaviour among the student population in Delft. One of the great successes has been the development of a highly accurate model that describes the intricate ritual of pub selection, that can be observed in the afternoon after the courses for the day are done.<\/p>\r\n\r\n<p>Students select a pub by casting votes. What makes it interesting from a sociological perspective is the mix of dominant behaviour by some students, and behaviour driven by peer pressure combined with randomness by other students.<\/p>\r\n\r\n<p>Speci\ufb01cally, the model used to describe the pub selection ritual is this:<\/p>\r\n\r\n<ul>\r\n\t<li>The dominant students, who have a strong preference for a particular pub, will (loudly) announce their vote. Several dominant students may vote for the same pub.<\/li>\r\n\t<li>Next, the remaining students vote, one by one. These non-dominant students vote probabilistically and are subject to peer pressure; the probability that they will vote for any speci\ufb01c pub is equal to the number of votes cast for that speci\ufb01c pub so far, divided by the total number of votes cast so far.<\/li>\r\n\t<li>Finally, when all votes are in, the pub with the highest number of votes is selected. If several pubs received the same highest number of votes, one of those is selected at random, with equal probability.<\/li>\r\n<\/ul>\r\n\r\n<p>For example, in one particular instance with seven students, the \ufb01ve dominant students start by declaring their votes for three different pubs. Three dominant students proclaim a preference for the \ufb01rst pub; the fourth dominant student announces her preference for a second pub, and the last dominant student votes for the third pub. This leaves the initial vote-count at (3, 1, 1).<\/p>\r\n\r\n<p>After that, the remaining two non-dominant students start to vote, one after the other. The first of them will pick either pub number one with probability 3\/5, pub number two with probability 1\/5, or pub number three with probability 1\/5. He happens to pick pub number three, and the vote-count becomes (3, 1, 2).<\/p>\r\n\r\n<p>Finally, the last student will pick either pub number one with probability 1\/2, pub number two with probability 1\/6, or pub number three with probability 1\/3. She, too, picks pub number three.<\/p>\r\n\r\n<p>This leaves the \ufb01nal vote-count at (3, 1, 3): a tie between pub one and pub three. This tie is broken by a coin-toss; with probability 1\/2, pub one is selected for that evening&rsquo;s drinking.<\/p>\r\n\r\n<p>You are a sociologist observing the ritual described above. After the dominant students are done declaring their votes, you want to know what the probabilities are for each pub that they will be serving drinks to the students that night.<\/p>\r\n","input":"<p>For each test case, the input consists of two lines:<\/p>\r\n\r\n<p>a line containing two positive integers: n, denoting the number of pubs in this trial (n &le; 5); and k, the total number of students, both dominant and non-dominant (k &le; 50).<br \/>\r\na line containing n positive integers (a<sub>1<\/sub>, a<sub>2<\/sub>, ..., a<sub>n<\/sub>), denoting the vote-count right after all dominant students have cast their vote.<\/p>\r\n\r\n<p>Furthermore, it is guaranteed that&nbsp;\\( k \\ge \\sum_{i=1}^{n}{a_i}\\)<\/p>\r\n","output":"<p>For each test case, write n lines of output containing the probability that the i&rsquo;th pub will be selected (1 &le; i &le; n), ordered by ascending i.<\/p>\r\n\r\n<p>Each line shall be of the form &lsquo;pub i: percentage %&rsquo;, where percentage is a \ufb02oating point number rounded to 2 digits after the decimal point.<\/p>\r\n\r\n<p>A space should separate the percentage number and the subsequent percent sign.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]

출처

ACM-ICPC > Regionals > Europe > Northwestern European Regional Contest > NWERC 2012 B번

  • 문제를 번역한 사람: baekjoon
  • 문제의 오타를 찾은 사람: corea