Answer: a) 0.01884167 b) 0.00017858 c) 5
Step-by-step explanation:
Given : The proportion of Californians own a car = 0.66
Sample size : n=8
We assume that each Californian is independent from other.
Let x be the number of Californians own a car.
Then, X [tex]\sim[/tex] Bin (n=8 , p=0.66)
Binomial probability formula = [tex]P(X=x)=^nC_xp^x(1-p)^x[/tex]
, where p=probability of getting success in each trial.
a) The probability that two of them own a car =
[tex]P(X=2)=^8C_2(0.66)^2(1-0.66)^6\\\\=\dfrac{8!}{2!6!}(0.66)^2(0.34)^6=0.01884167[/tex]
∴ The probability that two of them own a car is 0.01884167.
(b) The probability at least one of the owns a car =
[tex]P(X\geq1)=1-P(X<1)\\\\=1-P(X=0)\\\\=1-^8C_0(0.66)^0(0.34)^8=0.00017858[/tex]
∴ The probability at least one of the owns a car is 0.00017858.
(c) The expected number of Californians own a car = [tex]\mu=np[/tex]
[tex]=(8)(0.66)=5.28\approx5[/tex]
Hence, the expected number of Californians own a car = 5
