수학동아 - 폴리매스

주니어 폴리매스 문제

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[KAIST 과학영재교육연구원] k4. 불량품을 찾아라

수학동아 2018.11.01 18:03 조회 2209

페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

문제 다음과 같이 $X_{1}, X_{2}, \cdots, X_{n}$ 총 $\large n$ 개의 상품이 있고, $\large i$ 번째 상품이 불량일 확률이 $\large \frac{1}{2(i+1)}$ 일 때, 불량품의 개수가 짝수일 확률을 구하시오.

(단, $\large n$ 은 2보다 크거나 같으며 불량품이 0개인 경우도 짝수라고 생각한다.)

이전글 k3. 유리수의 정체를 밝혀라

다음글 k5. 명화 퍼즐 맞추기

문제 찜하기 목록보기

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21세기오일러 Lv.11 2018.11.01 18:07

폴리매스 문제는 일찍 나오는데 주니어 폴리매스는 늦게 나오네요.

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- 주니어멘토 Lv.1 2018.11.01 18:14
  
  안녕하세요 주니어폴리매스 멘토입니다!
  
  주니어폴리매스 문제는 매월 1일에 업로드되고 있습니다.
  
  감사합니다!
  
  좋아요1
21세기오일러 Lv.11 2018.11.01 18:55

불량품이 홀수일 확률을 구해서 전체확률에 빼면 되지 않을까요?

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21세기오일러 Lv.11 2018.11.01 19:08

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그러면 홀수일 확률은

$\sum_{x=1}^{\frac{n+1}{2}}4x$ 입니다.

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- 주니어멘토 Lv.1 2018.11.01 23:02
  
  안녕하세요 주니어폴리매스 멘토입니다!
  
  적어주신 합을 보면 첫 항만 생각해도 4이상의 값을 갖게 되는데,
  
  확률은 0과 1사이의 값이 되어야한다는 것을 고려해보시면 좋을 것 같습니다.
  
  문제가 '각 상품이 불량인지 아닌 지'에 대한 확률이 주어져있고, 결과적으로 전체 상품 중 불량품의 수가 짝수가 되는 사건의 확률을 구한다는 것을 잘 생각해보시길 바랍니다.
  
  감사합니다!
  
  좋아요0
21세기오일러 Lv.11 2018.11.02 02:23

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제가 실수 했었네요.

n이 짝수일때 짝수번째가불량인 확률은 $\sum_{x=1}^{\frac{n+1}{2}}\frac{1}{4x+2}$ 입니다.

n이 홀수일때 짝수번째가 불량인 확률은 $\sum_{x=1}^{\frac{n-1}{2}}\frac{1}{4x+2}$ 입니다.

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- 21세기오일러 Lv.11 2018.11.02 02:24
  
  이제 여기까지는 맞나요?
  
  좋아요0
- 주니어멘토 Lv.1 2018.11.02 02:53
  
  원하시는 표현은 아마 4x+4가 되고, 짝수일 때 시그마의 위끝도 n/2가 되어야할 것입니다. 그렇게 수정하시면 적으신 것은 맞습니다. (짝수번째 중 적어도 하나가 불량일 확률이 되겠죠)
  하지만, 문제에서 원하는 것은 짝수번째냐 아니냐와는 무관하게 불량품의 총 개수의 홀짝성에 관한 문제라는 것을 유의하셔야 합니다!
  
  다른 분들도 보시고 함께 논의가 진행되면 가장 좋을 것 같으니 천천히 생각해보시면 될 것 같습니다.
  
  감사합니다!
  
  좋아요1
21세기오일러 Lv.11 2018.11.02 02:36

생각보다 힘드네요.

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구머 Lv.6 2018.11.03 04:53

점화식을 사용해 봅시다^^

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EulerD Lv.1 2018.11.11 02:57

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생각보다 많은 분들이 고전하시는 부분이,

답을 직접 찾아나가는 대부분의 확률 문제와는 달리

이 문제는 정수 n 에 관한 문제이므로 수학적 귀납법을 사용하면 좋습니다.

sol.

정수 n 에 대하여 불량품이 짝수개일 확률을 A_n이라 합시다.

자명하게 불량품이 홀수개일 확률은 (1-A_n)입니다.

이때, 다음 관찰을 해봅시다,

n개일때 불량이 짝수개이려면,

1. 첫번째부터 n-1번째까지 불량품이 짝수개이고, n번째가 불량품이거나,

2. ,, ,, ,, 홀수개 ,, ,, 불량품이 아니거나

입니다.

따라서, $A_{n} = A_{n-1}(1-\frac{1}{2(n+1)}) + (1-A_{n})\frac{1}{2(n+1)}$ 임을 알수 있습니다.

A₁ 은 3/4이므로, 약간의 노가다를 통해 $\tiny A_n= \frac{n+2}{2n+2}$ 임을 알수 있습니다.

이는 수학적 귀납법을 통해 증명 가능합니다. ( 간단하므로 생략 )

$\therefore \frac{n+2}{2n+2}$

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- EulerD Lv.1 2018.11.11 02:57
  
  맞는지는 모르겠지만.. 푼듯해요..
  
  약간의 노가다후 증명함.
  
  좋아요1
- 바람개비 Lv.5 2018.11.17 09:57
  
  와~
  
  점화식을 활용한다는 생각을 못했네요... 대단하십니다!
  
  실례지만 점화식을 이용하는 것 같은데 해결이 안되는 문제 하나 풀어 주실 수 있으세요?
  
  좋아요1
- EulerD Lv.1 2018.11.18 07:37
  
  저야 좋지요~
  
  좋아요0
- 바람개비 Lv.5 2018.11.18 08:28
  
  잘됬네요:)
  
  여기 링크 겁니다.
  
  좋아요0
베네딕트0724 Lv.6 2018.11.21 08:02

풀린 건가요? 놀랍군요!

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