Re: [機統] 硬幣問題
※ 引述《celestialgod (攸藍)》之銘言：
: A coin, having probability p of landing heads, is continually flipped until
: at least two head and one tail have been flipped. Find the expected number od
: flips needed.
: 推 Aweather :窮舉就可以了 01/26 18:34
: 窮舉?可以簡述一下過程嗎?
假設共需擲 X 次
P[X=3] = 3p^2(1-p) (3次中2正1反)
當n>=4, P[X=n] = C(n-1,1) p^2(1-p)^(n-2) (第2次正面出現在第n次)
2/p = 2 C(1,1) p^2 + 3 C(2,1) p^2 (1-p) + 4 C(3,1) p^2 (1-p)^2 +
5 C(4,1)p^2(1-p)^3 + ..... (Pascal(p,2)的期望值)
E[X] = 3*3p^2(1-p) + 4 C(3,1) p^2 (1-p)^2 + 5 C(4,1)p^2(1-p)^3 + ...
= (2 C(1,1) p^2 + 3 C(2,1) p^2 (1-p) + 4 C(3,1) p^2 (1-p)^2 +...)
+ 3*3p^2(1-p) - 2 C(1,1) p^2 - 3 C(2,1) p^2 (1-p)
= 2/p - 2p^2 + 3p^2(1-p)
= 2/p + p^2(1-3p)
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推
01/28 14:13, , 1F
01/28 14:13, 1F
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