시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
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문제

이제 여행을 끝나고 돌아갈 날이 되었다. 물론 아쉬운 것은 많지만 그 중 가장 아쉬운 것은 내가 2만 피트 상공을 날아가는 동안 윔블던 테니스 시합이 끝난다는 점이다.

어, 잠깐. 잠시 잘못 생각했다. 시합이 시작하는 거야 물론 낮이지만, 테니스의 특성상 대회가 충분히 길어질 수 있다. 만일 저녁까지 시합이 계속된다면 아마 대회를 볼 수 있을 텐데. 언제쯤 시합이 끝나려나?

난 모든 선수들에 대해 서브권을 가졌을 때 그 선수가 어느 정도의 확률로 이기는지 모두 파악하고 있다. 다만 테니스 시합은 여러 세트로 이루어져 있고, 그 세트는 각각 여러 게임으로 이루어져 있으며, 누가 서브를 하냐에 따라 승률이 다르기 때문에 나 혼자는 대회가 얼마나 길어질지 예측할 수가 없다.

테니스 대회의 규칙은 간단하다.

위에 말했듯이 어떤 두 선수간의 경기는 여러 세트로 이루어져 있다.

만일 어떤 선수가 먼저 3세트를 승리한다면 곧바로 경기는 종료된다.

따라서, 어떤 경기의 최종 세트 스코어는 반드시 3-0, 3-1, 3-2 중 하나이다.

그리고 각 세트는 여러 게임으로 이루어져 있다.

각 게임은 반드시 5분씩 진행되며 만일 어떤 선수가

  • 6게임 이상 승리
  • 상대 선수보다 2게임 이상 더 이김

을 만족한다면 한 세트가 끝난다. 이때 세트의 승자는 6게임 이상 승리한 그 선수가 된다.

만일 각 세트에서 6-6의 게임 스코어가 나온다면, Tie-Break 게임 하나를 더 진행하여 승자를 결정한다.

따라서, 각 세트에서 나올 수 있는 게임 스코어는 6-0, 6-1, 6-2, 6-3, 6-4, 7-5, 그리고 Tie-Break를 통한 결과로 7-6까지 총 7가지밖에 없다.

각 게임에서는 한 선수가 게임 내내 서브를 하며, 게임이 끝날 때마다 서브권은 변경된다. 서브권은 승패와 관계없이 게임이 끝날 때마다 교대로 갖게 되며, 이 규칙은 세트 단위를 넘어 경기 내내 적용된다. 예를 들어, A와 B의 경기에서 A가 먼저 서브를 하고 게임 스코어 6-1로 한 세트를 승리로 끝냈다면, 다음 세트에서는 B가 먼저 서브를 한다. 물론 이것은 A가 먼저 서브를 하고 6-1의 게임 스코어로 B가 한 세트를 승리했을 경우에도 적용되어 다음 세트에서는 B가 먼저 서브를 한다. 즉, 이전 세트의 마지막 게임에서 서브를 한 선수의 상대 선수가 이번 세트의 첫 서브를 하게 된다는 것이다.

내가 아는 정보는 어느 두 선수가 만났을 때 서브권을 가진 선수가 어느 정도의 확률로 이기는지에 대한 정보 뿐이다.

그래서 이 정보만으로 두 선수간의 경기가 최종 종료되기까지 걸리는 기대시간을 계산해줄 프로그램이 필요하다.

입력

입력의 첫 줄엔 테스트 케이스의 수 T가 주어진다.

각 테스트 케이스는 두 줄로 이루어져 있다.

첫 줄엔 처음으로 서브를 하는 선수의 성과 이름이 공백으로 구분되어 주어지며, 그에 이어 그 선수가 서브권을 가진 게임에서 승리할 확률이 백분율의 100배인 정수로 주어진다.

그리고 다음 줄엔 첫 줄의 선수와 경기를 하는 선수의 성과 이름이 공백으로 구분되어 주어지며, 역시 이 선수가 서브권을 가진 게임에서 어느 정도의 확률로 승리하는지가 주어진다.

예를 들어, 예제의 첫 테스트 케이스를 살펴보면

Rafael Nadal 50

Roger Federer 50

이 테스트 케이스는

첫 서브를 하는 Rafael Nadal은 서브권을 가졌을 때 50%의 확률로 승리하며, Roger Federer 선수도 서브권을 가졌을 때 50%의 확률로 승리한다는 것이다.

출력

각 테스트 케이스에 대해, 주어진 두 선수의 경기가 얼마나 진행될지 소수점 7번째 자리에서 반올림하여 여섯 번째 자리까지 출력한다. 절대/상대 오차는 10-6까지 허용한다.

예제 입력 1

2
Rafael Nadal 50
Roger Federer 50
Pete Sampras 100
Hamza Darwish 0

예제 출력 1

Case #1: 199.281006
Case #2: 90.000000

힌트

두 번째 테스트 케이스에서는 Pete Sampras가 모든 게임을 이기며 6-0의 게임 스코어로 묶인 세트 3개를 연달아 승리하여 총 18경기만을 하게 되며, 총 경기는 항상 90분만에 끝나게 된다.

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\ud544\uc694\ud558\ub2e4.<\/p>\r\n","input":"<p>\uc785\ub825\uc758 \uccab \uc904\uc5d4 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uc218 T\uac00 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n\r\n<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub294 \ub450 \uc904\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4.<\/p>\r\n\r\n<p>\uccab \uc904\uc5d4 \ucc98\uc74c\uc73c\ub85c \uc11c\ube0c\ub97c \ud558\ub294&nbsp;\uc120\uc218\uc758 \uc131\uacfc \uc774\ub984\uc774 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc8fc\uc5b4\uc9c0\uba70, \uadf8\uc5d0 \uc774\uc5b4 \uadf8 \uc120\uc218\uac00 \uc11c\ube0c\uad8c\uc744 \uac00\uc9c4 \uac8c\uc784\uc5d0\uc11c \uc2b9\ub9ac\ud560 \ud655\ub960\uc774 \ubc31\ubd84\uc728\uc758 100\ubc30\uc778 \uc815\uc218\ub85c&nbsp;\uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n\r\n<p>\uadf8\ub9ac\uace0 \ub2e4\uc74c \uc904\uc5d4 \uccab \uc904\uc758 \uc120\uc218\uc640 \uacbd\uae30\ub97c \ud558\ub294 \uc120\uc218\uc758 \uc131\uacfc \uc774\ub984\uc774 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc8fc\uc5b4\uc9c0\uba70, \uc5ed\uc2dc \uc774 \uc120\uc218\uac00 \uc11c\ube0c\uad8c\uc744 \uac00\uc9c4 \uac8c\uc784\uc5d0\uc11c \uc5b4\ub290 \uc815\ub3c4\uc758 \ud655\ub960\ub85c \uc2b9\ub9ac\ud558\ub294\uc9c0\uac00 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n\r\n<p>\uc608\ub97c \ub4e4\uc5b4, \uc608\uc81c\uc758 \uccab \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub97c \uc0b4\ud3b4\ubcf4\uba74<\/p>\r\n\r\n<p>Rafael Nadal 50<\/p>\r\n\r\n<p>Roger Federer 50<\/p>\r\n\r\n<p>\uc774 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub294<\/p>\r\n\r\n<p>\uccab \uc11c\ube0c\ub97c \ud558\ub294&nbsp;Rafael Nadal\uc740 \uc11c\ube0c\uad8c\uc744 \uac00\uc84c\uc744 \ub54c 50%\uc758 \ud655\ub960\ub85c \uc2b9\ub9ac\ud558\uba70, Roger Federer \uc120\uc218\ub3c4 \uc11c\ube0c\uad8c\uc744 \uac00\uc84c\uc744 \ub54c 50%\uc758 \ud655\ub960\ub85c \uc2b9\ub9ac\ud55c\ub2e4\ub294 \uac83\uc774\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc5d0 \ub300\ud574, \uc8fc\uc5b4\uc9c4 \ub450 \uc120\uc218\uc758 \uacbd\uae30\uac00 \uc5bc\ub9c8\ub098 \uc9c4\ud589\ub420\uc9c0 \uc18c\uc218\uc810 7\ubc88\uc9f8 \uc790\ub9ac\uc5d0\uc11c \ubc18\uc62c\ub9bc\ud558\uc5ec \uc5ec\uc12f \ubc88\uc9f8 \uc790\ub9ac\uae4c\uc9c0&nbsp;\ucd9c\ub825\ud55c\ub2e4. \uc808\ub300\/\uc0c1\ub300 \uc624\ucc28\ub294 10<sup>-6<\/sup>\uae4c\uc9c0 \ud5c8\uc6a9\ud55c\ub2e4.<\/p>\r\n","hint":"<p>\ub450 \ubc88\uc9f8 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc5d0\uc11c\ub294 Pete Sampras\uac00 \ubaa8\ub4e0 \uac8c\uc784\uc744 \uc774\uae30\uba70 6-0\uc758 \uac8c\uc784 \uc2a4\ucf54\uc5b4\ub85c \ubb36\uc778 \uc138\ud2b8 3\uac1c\ub97c \uc5f0\ub2ec\uc544 \uc2b9\ub9ac\ud558\uc5ec \ucd1d 18\uacbd\uae30\ub9cc\uc744 \ud558\uac8c \ub418\uba70, \ucd1d \uacbd\uae30\ub294&nbsp;\ud56d\uc0c1 90\ubd84\ub9cc\uc5d0 \ub05d\ub098\uac8c \ub41c\ub2e4.<\/p>\r\n","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"9360","problem_lang":"1","title":"Wimbledon","description":"<p>I&rsquo;m flying back to Cairo tomorrow, and do you know what is sad about that? Well, in addition to leaving beautiful Lebanon? Yep, I&rsquo;m going to miss the 2011 Wimbledon final match which is going to happen while I&rsquo;m 20,000 feet above sea level!&nbsp;<\/p>\r\n\r\n<p>But hey, wait! I may have a shot here! At Wimbledon, matches are played only at day light, so maybe if players were not ready to give up easily on each point, then the match will extend until sunset, and the match would be suspended and resumed on Monday just for my convenience! So what I need to know is how long on average a game between two players will last, and to help me feel better you should write a computer program to determine that!&nbsp;<\/p>\r\n\r\n<p>The result of a tennis match is determined by the number of &ldquo;sets&rdquo; each player wins, the first player to win three sets wins the match. Accordingly, all possible match results are 3-0, 3-1 and 3-2. The result of each set is determined by the number of &ldquo;games&rdquo; in the set each player wins, the first player to win six or more games with two or more game difference from the opponent wins the set, however if the result of a set (including the last set) leveled at 6-6 then a tie-break game is played to determine the set winner, Accordingly all possible set results are 6-0, 6-1, 6-2, 6-3, 6-4, 7-5 and 7-6 using a tie breaker game!&nbsp;<\/p>\r\n\r\n<p>During each game (including tie-break), one player has the serve, this means that this player must start the play for all points of the game by hitting the ball, serving the ball is considered a big advantage. Assume that the serve starts at the first player and then alternates after each game till the end.&nbsp;<\/p>\r\n\r\n<p>Given the probability of winning a game on serve for each player against his opponent, and assuming that each game lasts for five minutes, calculate the expected duration of the match.<\/p>\r\n","input":"<p>The first line of input contains an integer T, the number of test cases. The first line of each test case contains the first and last names of the first player, followed by an integer (0 &lt;= A &lt;= 100) where A\/100 is the probability that the first player wins a game on his serve against the second player. The second line contains the first and last names of the second player, followed by an integer (0 &lt;= B &lt;= 100) where B\/100 is the probability that the second player wins a game on his serve against the first player.&nbsp;<\/p>\r\n","output":"<p>For each test case, print the case number followed by the expected duration of the match in minutes rounded to six decimal digits.&nbsp;<\/p>\r\n","hint":"<p>In the second test case, I don&rsquo;t stand a chance against Pete! He always wins all games on his serve (probability 100\/100) and he also always wins all games on my serve (as I win with probability 0\/100), so he always wins the match with three straight sets (6-0, 6-0, 6-0), a total of 18 games played, each lasting five minutes for a total of 90 minutes match.<\/p>\r\n","original":"1","problem_lang_code":"\uc601\uc5b4"}]