Answer:
0.618
Step-by-step explanation:
This is a binomial distribution. We are to determine the probability of people unsatisfied. If we denote the probability of being satisfied with [tex]q[/tex], then [tex]q=0.8[/tex]. Let [tex]p[/tex] be the probability of being unsatisfied. Then [tex]p+q=1[/tex]
[tex]p=1-q =1-0.8=0.2[/tex]
Since the random sample is 20 and [tex]p[/tex] is small enough, we can use the Poisson probability function
[tex]P(X=x) = \dfrac{e^{-\lambda}\lambda^x}{x!}[/tex]
where [tex]\lambda[/tex] is the mean and is given by [tex]\lambda=np=20\times0.2=4[/tex].
Denoting the probability of at most 4 unsatisfied with [tex]P(x\le4)[/tex],
[tex]P(x\le4)=P(x=0)+P(x=1)+P(x=2)+P
(x=3)+P(x=4)[/tex]
[tex]P(x=0)= \dfrac{e^{-4}4^0}{0!} = e^{-4}[/tex]
[tex]P(x=1)= \dfrac{e^{-4}4^1}{1!} = 4e^{-4}[/tex]
[tex]P(x=2)= \dfrac{e^{-4}4^2}{2!} = 8e^{-4}[/tex]
[tex]P(x=3)= \dfrac{e^{-4}4^3}{3!} = e^{-4}\dfrac{32}{3}[/tex]
[tex]P(x=4)= \dfrac{e^{-4}4^4}{4!} = e^{-4}\dfrac{32}{3}[/tex]
[tex]P(x\le4)=e^{-4}(1+4+8+\dfrac{32}{3}+\dfrac{32}{3})=0.018\times34.33 = 0.618[/tex]
