Re: [理工] [機率]請問一題

[KAINTS (大安Eason哥)]

※ 引述《z3992270 (Eazon)》之銘言：
: 請問各位高手這題
: There are k+1 coins in a box.
: The ith coin will,when flipped,turn up head withprobability i/k
: ,i=0,1,...,k . A coin is randomly slected from the box and is then
: repeatedly filpped.
: If the first n flips all results in heads, what is the conditional
: probability that the (n+1)st flip will do likewise?
: 答案寫
: P((n+1)st head | n次head)
:  1   1  2       k-1 k    k+1
: =--(--+--+.....+---+--)=----    (*)
:  k   k  k        k  k    2k
     {coin0 coin1 coin2 ... coink-1 coink }
prob   0/k    1/k    2/k      k-1/k   k/k
(head)

first,we need all results in heads,so we can't choose coin0
then P(n次head)=1/k (total k's coins,except coin )

second ,we have to choose (n+1)st head is head
Prob=1/k+2/k+...k/k

so the answer is  (*)
: 我自己是把n+1次正面的機率除於n次正面的機率
: 就等於
:  k             k
: Σ(i/k)^(n+1)/Σ(j/k)^n
: i=0           j=0
: 不知道這樣對不對??
: 請各位大大指點一下  謝謝

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推 z3992270:可是在所有head結果當中不是選擇應該看coin是什麼而機率
→ z3992270:有所變化嗎?我覺得應該不是1/k
→ z3992270:我覺得應該是(i/k)^n/Σ(j/k)^n
→ KAINTS:錯...那是正面的機率！1/k是指從這k個會出現正面的硬幣挑一
→ KAINTS:個的機率，你搞錯了。
推 z3992270:所以說他是從出現n次正面的硬幣中又隨機挑出一個算他出現
→ z3992270:的機率囉!?