[機統] 取球期望值
題為
甕中有三紅球,七白球,每次從中取一球,若取到紅球則加入一個白球,
若取到白球則不放入任何顏色之球,求取到3紅球的期望值
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對這題的感覺,
猜想應該試問次數期望值
以隨機變數X為取球數,3<=X<=12
P(X=3)=(3/10)*(2/10)*(1/10)
P(X=4)
=P(wrrr)+P(rwrr)+P(rrwr)
=(7/10)*(3/9)*(2/9)*(1/9)+(3/10)(7/10)(2/9)(1/9)+(3/110)(2/10)(7/10)(1/9)
=(7*3!)[1/(10*9^3)+1/(10^2*9^2)+1/(10^3*9)
P(X=5)=P(wwrrr)+....+P(rrwwr)=....
.
.
.
P(X=12)=P(wwwwwwwrrr)+...+P(rrwwwwwwwr)
這是我腦中的土法煉鋼暴力法,這慢慢加有點不得了@@
有請好手們不吝分享其他切入觀點 ! 謝謝
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