另法:
1. X=<9的機率
A袋: 3個紅球/total 13個球
B袋: 空的
P(X<=9)
=從A袋中取出9球,取後放B袋,所有的紅球皆取到B袋的機率
=(9!/3!6!)/(13!/3!10!)
=42/143
2.同理 P(X<=8)
=(8!/3!5!)/(13!/3!10!)
=C(8,3)/C(13,3)
=28/143
so,
P(X=9)=P(X<=9)-P(X<=8)
=(C(9,3)-C(8,3))/C(13,3)
=C(8,2)/C(13,3)
=14/143
3.X的期望值
=1*P(x=1)+2*P(x=2)+3*P(x=3)+...+13*P(x=13)
=(1*0+2*0+3*C(2,2)+4*C(3,2)+5*C(4,2)+...+13*C(12,2) )/C(13,3)
=3*(C(3,3)+C(4,3)+C(5,3)+...+C(13,3))/C(13,3)
=3*C(14,4)/C(13,3)
=10.5
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