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~ MC 及格概率 ~

某考試共有 30 道選擇題, 每題共 4 個選項, 考生須選擇正確的一個,

答對 16 道題或以上為及格。

考生 A 只懂得其中 5 道題 , 有 12 道題只能排除兩個選項 ,

有 8 道題只能排除一個選項 , 而最後 5 道題則全沒頭緒。若考生 A 會於其認為可能的答案中隨意選一個,問考生 A 於該試之及格概率為何?

5 個解答

評分
  • 6 年前

    6.77%(三位有效數字)

    你有冇正確答案?

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  • futari
    Lv 5
    7 年前

    Let the probability that the student answers the questions that s/he knows be a. Then a=1,

    Let the probability that the student answers the questions that s/he could eliminate 2 incorrect options be b. Then b=1/2.

    Let the probability that the student answers the questions that s/he could eliminate 1 incorrect options be c. Then c=1/3.

    Let the probability that the student answers the questions that s/he could not eliminate any incorrect options be d. Then d=1/4.

    Denote C(n, r) as the binomial coefficient nCr

    Of the 4 categories of probabilities of answering, let the corresponding number of correct questions be A', B', C', D'. Then 16<= (A'+B'+C'+D') <= 30 and A' belongs to [0, 5], B' belongs to [0, 8], C' belongs to [0,12], and D' belongs to [0, 5].

    Let the total number of correct question be K.

    K=A'+B'+C'+D' ==> D'=K-A'-B'-C'

    The probability P(K) of answering K correct questions

    = C(K, A') * C(5, A')a

    * C(K-A', B') * C(12, B')b

    * C(K-A'-B',C') * C(8, C')c

    * C(K-A'-B'-C', K-A'-B'-C') * C(5, K-A'-B'-C')d

    The required probability is

    = Summation from K=16 to K=30 of P(K).

    資料來源: Me
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  • 7 年前

    答對 5 道懂得的題目 (A組) 的概率 = 100%

    答對 12 道排除兩個選項的題目 (B組) 的概率 = 50%

    答對 8 道排除一個選項的題目 (C組) 的概率 ~ 33%

    答對 5 道沒有頭緒的題目 (D組) 的概率 = 25%

    答對 30 條的概率

    = (50%)^12 x (33%)^8 x (25%)^5

    答對 29 條的概率

    = 11C1 x (50%)^11 x (33%)^8 x (25%)^5 x (50%)^1 11B 8C 5D

    + 8C1 x (50%)^12 x (33%)^7 x (25%)^5 x (67%)^1 12B 7C 5D

    + 5C1 x (50%)^12 x (33%)^8 x (25%)^4 x (75%)^1 12B 8C 4D

    答對 28 條的概率

    = 12C2 x (50%)^10 x (33%)^8 x (25%)^5 x (50%)^2 10B 8C 5D

    + 8C2 x (50%)^12 x (33%)^6 x (25%)^5 x (67%)^2 12B 6C 5D

    + 5C3 x (50%)^12 x (33%)^8 x (25%)^3 x (75%)^2 12B 8C 3D

    + 11C1 x 8C1 x (50%)^11 x (33%)^7 x (25%)^5 x (50%)^1 x (67%)^1 11B 7C 5D

    + 11C1 x 5C1 x (50%)^11 x (33%)^8 x (25%)^4 x (50%)^1 x (75%)^1 11B 8C 4D

    + 8C1 x 5C1 x (50%)^12 x (33%)^7 x (25%)^4 x (67%)^1 x (75%)^1 12B 7C 4D

    ...

    這樣計算下去只有一個下場:傻左……

    資料來源: knowledge
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  • CK
    Lv 7
    7 年前

    少了

    [ 答對 16 道題或以上為及格。 ]

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  • 7 年前

    懂 : 5/30

    只能排除兩個選項: 12/30, 選擇正確: 1/2

    只能排除一個選項: 8/30, 選擇正確: 1/3

    全沒頭緒: 5/30, 選擇正確: 1/4

    及格概率: [5+12(1/2)+8(1/3)+5(1/4)]/30 = 0.4972 = 49.72%

    資料來源: myself
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