Answer:
The probability is 0.4207
Step-by-step explanation:
The probability of a home-based computer having access to on-line services is p = 0.2 (data from the exercise)
Then, the probability of a home-based computer not having access to on-line services is p = 1 - 0.2 = 0.8
We are going to use this probability (p = 0.8) to solve the exercise.
Let's define the random variable X
X : ''Number of home-based computers not having access to on-line services''
X can be modeled as a binomial random variable
X ~ Bi(p,n)
X ~Bi(0.8,25)
Where p is the success probability and n is the number of Bernoulli independent experiments we are taking place.
We are going to count ''a success'' as a computer not having access to on-line services.
The binomial probability function is :
![P(X=x)=(nCx)p^{x}(1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%3Dx%29%3D%28nCx%29p%5E%7Bx%7D%281-p%29%5E%7Bn-x%7D)
Where P(X=x) is the probability of the random variable X to assume the value x
nCx is the combinatorial number define as
![nCx=\frac{n!}{x!(n-x)!}](https://tex.z-dn.net/?f=nCx%3D%5Cfrac%7Bn%21%7D%7Bx%21%28n-x%29%21%7D)
p is the success probability and n the number of Bernoulli independent experiments taking place.
In our exercise,
![p=0.8\\n=25](https://tex.z-dn.net/?f=p%3D0.8%5C%5Cn%3D25)
We are looking for :
![P(X>20)=P(X=21)+P(X=22)+P(X=23)+P(X=24)+P(X=25)](https://tex.z-dn.net/?f=P%28X%3E20%29%3DP%28X%3D21%29%2BP%28X%3D22%29%2BP%28X%3D23%29%2BP%28X%3D24%29%2BP%28X%3D25%29)
![P(X>20)=(25C21)0.8^{21}0.2^{4}+(25C22)0.8^{22}0.2^{3}+(25C23)0.8^{23}0.2^{2}+(25C24)0.8^{24}0.2^{1}+(25C25)0.8^{25}0.2^{0}](https://tex.z-dn.net/?f=P%28X%3E20%29%3D%2825C21%290.8%5E%7B21%7D0.2%5E%7B4%7D%2B%2825C22%290.8%5E%7B22%7D0.2%5E%7B3%7D%2B%2825C23%290.8%5E%7B23%7D0.2%5E%7B2%7D%2B%2825C24%290.8%5E%7B24%7D0.2%5E%7B1%7D%2B%2825C25%290.8%5E%7B25%7D0.2%5E%7B0%7D)
![P(X>20)=0.1867+0.1358+0.0708+0.0236+0.8^{25}](https://tex.z-dn.net/?f=P%28X%3E20%29%3D0.1867%2B0.1358%2B0.0708%2B0.0236%2B0.8%5E%7B25%7D)
![P(X>20)=0.4207](https://tex.z-dn.net/?f=P%28X%3E20%29%3D0.4207)
Finally, the probability of finding that more than 20 of 25 home-based computers do not have access to on-line services is 0.4207