~ MC 及格概率 ~

6.77%(三位有效數字)
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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).

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……

少了

[ 答對 16 道題或以上為及格。 ]

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%