請問一題數學機率問題

答案是： (A)、(C)

H : 該賓客誠實（不是竊賊）
H' : 該賓客說謊（是竊賊）
C : 測謊器顯示該賓客誠實
C' : 測謊器顯示該賓客說謊

P(H) = 49/50
P(H') = 1/50

P(C'|H') = 99%
P(C' and H') / P(H') = 99%
P(C' and H') / (1/50) = 99%
P(C' and H') = 99% x (1/50) = 0.0198

P(C|H') = 1 - 99% = 1%
P(C and H') / P(H') = 1%
P(C and H') / (1/50) = 1%
P(C and H') = 1% x (1/50) = 0.0002

P(C|H) = 90%
P(C and H) / P(H) = 90%
P(C and H) / (49/50) = 90%
P(C and H) = 90% x (49/50) = 0.882

P(C'|H) = 1 - 90% = 10%
P(C' and H) / P(H) = 10%
P(C' and H) / (49/50) = 10%
P(C' and H) = 10% x (49/50) = 0.098

(A) 正確
P(測謊器顯示賓客說謊)
= P(C')
= P(C' and H') + P(C' and H)
= 0.0198 + 0.098
= 0.1178 > 10%

(B) 錯誤
P(測謊器顯示一賓客說謊，而該賓客正是竊賊)
= P(H'|C')
= P(C' and H') / P(C')
= 0.0198/0.1178
≈ 0.168 < 50%

(C) 正確
P(測謊器顯示一賓客誠實，而該賓客卻是竊賊)
= P(H'|C)
= P(C and H') / P(C)
= 0.0002 / (1 - 0.1178)
≈ 0.000225 < 20%

(D) 錯誤
賓客數目愈多，P(H) 愈多，P(H') 愈少。
機率的計算涉及 P(H) 和 P(H')，故此人數的多少會改變機率的大小。

(E) 錯誤
P(測謊機顯示所有賓客均未說謊的機率)
= [P(C and H)]^49 + P(C and H') > 0

為甚ㄇ我看不到100000000000

http://tw.myblog.yahoo.com/sincos-heart/article?mid=-2&next=2289&l=a&fid=45
第6題