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 :
[tex]P(X=x)=(nCx)p^{x}(1-p)^{n-x}[/tex]
Where P(X=x) is the probability of the random variable X to assume the value x
nCx is the combinatorial number define as
[tex]nCx=\frac{n!}{x!(n-x)!}[/tex]
p is the success probability and n the number of Bernoulli independent experiments taking place.
In our exercise,
[tex]p=0.8\\n=25[/tex]
We are looking for :
[tex]P(X>20)=P(X=21)+P(X=22)+P(X=23)+P(X=24)+P(X=25)[/tex]
[tex]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}[/tex]
[tex]P(X>20)=0.1867+0.1358+0.0708+0.0236+0.8^{25}[/tex]
[tex]P(X>20)=0.4207[/tex]
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
