Respuesta :
Answer:
a) There is a 2.43% probability that exactly 8 of the 20 Internet browser users use Chrome as their Internet browser.
b) There is an 80.50% probability that at least 3 of the 20 Internet browsers users use Chrome as their Internet browser.
c) The expected number of Chrome users is 4.074.
d) The variance for the number of Chrome users is 3.2441.
The standard deviation for the number of Chrome users is 1.8011.
Step-by-step explanation:
For each Internet browser user, there are only two possible outcomes. Either they use Chrome, or they do not. This means that we can solve this problem using concepts of the binomial probability distribution.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinatios of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
In this problem we have that:
Google Chrome has a 20.37% share of the browser market. This means that [tex]p = 0.2037[/tex]
20 Internet users are sampled, so [tex]n = 20[/tex].
a.Compute the probability that exactly 8 of the 20 Internet browser users use Chrome as their Internet browser.
This is P(X = 8).
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 8) = C_{20,8}.(0.2037)^{8}.(0.7963)^{12} = 0.0243[/tex]
There is a 2.43% probability that exactly 8 of the 20 Internet browser users use Chrome as their Internet browser.
b.Compute the probability that at least 3 of the 20 Internet browsers users use Chrome as their Internet browser.
Either there are less than 3 Chrome users, or there are three or more. The sum of the probabilities of these events is decimal 1. So:
[tex]P(X < 3) + P(X \geq 3) = 1[/tex]
[tex]P(X \geq 3) = 1 - P(X < 3)[/tex]
In which
[tex]P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)[/tex]
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{20,0}.(0.2037)^{0}.(0.7963)^{20} = 0.0105[/tex]
[tex]P(X = 1) = C_{20,1}.(0.2037)^{1}.(0.7963)^{19} = 0.0538[/tex]
[tex]P(X = 2) = C_{20,2}.(0.2037)^{2}.(0.7963)^{18} = 0.1307[/tex]
[tex]P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) = 0.0105 + 0.0538 + 0.1307 = 0.1950[/tex]
[tex]P(X \geq 3) = 1 - P(X < 3) = 1 - 0.1950 = 0.8050[/tex]
There is an 80.50% probability that at least 3 of the 20 Internet browsers users use Chrome as their Internet browser.
c.For the sample of 20 Internet browser users, compute the expected number of Chrome users
We have that, for a binomial experiment:
[tex]E(X) = np[/tex]
So
[tex]E(X) = 20*0.2037 = 4.074[/tex]
The expected number of Chrome users is 4.074.
d.For the sample of 20 Internet browser users, compute the variance and standard deviation for the number of Chrome users.
We have that, for a binomial experiment, the variance is
[tex]Var(X) = np(1-p)[/tex]
So
[tex]Var(X) = 20*0.2037*(0.7963) = 3.2441[/tex]
The variance for the number of Chrome users is 3.2441.
The standard deviation is the square root of the variance. So
[tex]\sqrt{Var(X)} = \sqrt{3.2441} = 1.8011[/tex]
The standard deviation for the number of Chrome users is 1.8011.
