機率一題
如下圖回復 1# nanpolend 的帖子
咦! 超過一個月的文?!假設樂透總共有N張，其中24張是有獎的
Notation: A (Ann), B(Ben), C(Cal), c(car), b(bicycle), w(Watch), R(prize-winner)
1. 即求\(\displaystyle P(A=c,B=b\ or\ w|A\in{R}, B\in{R}, C\in{R})\)
\(\displaystyle =\frac{P(A=c, B=b\ or\ w, C\in{R})}{P(A\in{R}, B\in{R}, C\in{R})}\)
\(\displaystyle =\frac{P(C\in{R}|A=c, B=b\ or\ w)P(A=c|B=b\ or\ w)P(B=b\ or\ w)}{\frac{24\cdot23\cdot22}{N(N-1)(N-2)}}\)
\(\displaystyle =\frac{\frac{22}{N-2}\cdot\frac{4}{N-1}\cdot\frac{20}{N}}{\frac{24\cdot23\cdot22}{N(N-1)(N-2)}}=\frac{10}{69}\)
2. (a) 即求\(\displaystyle P(A=c, B=c|C=c, A\in{R}, B\in{R}, C\in{R})\)
\(\displaystyle =\frac{P(A=c, B=c, C=c)}{P(A\in{R}, B\in{R}, C=c)}\)
\(\displaystyle =\frac{\frac{4\cdot3\cdot2}{N(N-1)(N-2)}}{P(A\in{R}|B\in{R}, C=c)P(B\in{R}|C=c)P(C=c)}\)
\(\displaystyle =\frac{\frac{4\cdot3\cdot2}{N(N-1)(N-2)}}{\frac{22}{N-2}\cdot\frac{23}{N-1}\cdot\frac{4}{N}}=\frac{3}{253}\)
(b)(c)(d)應該都是類似解法吧XD
[[i] 本帖最後由 Pacers31 於 2013-7-29 12:30 PM 編輯 [/i]]
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