Let X= the number of grey cars: X∼Bin(n,p).
(a) Given n=10,p=0.2,q=1−p=0.8
P(X≥5)=1−P(X=0)−P(X=1)
−P(X=2)−P(X=3)−P(X=4)
=1−(010)(0.2)0(0.8)10−0−(110)(0.2)1(0.8)10−1
−(210)(0.2)2(0.8)10−2−(310)(0.2)3(0.8)10−3)
−(410)(0.2)4(0.8)10−4=0.0327934976
(b) Given n=5,p=0.2,q=1−p=0.8
P(X=0)=(05)(0.2)0(0.8)5−0=0.32768
(c) Given n=25,p=0.2,q=1−p=0.8
E[X]=np=25(0.2)=5
Var[X]=npq=25(0.2)(0.8)=4
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