Respuesta :
Answer:
The probability of getting B=40 is [tex]0.11\times 10^{-11}[/tex] which is negligible.
Step-by-step explanation:
Given that 31 percent of the residents of a certain state who are age 25 years or older have a bachelor’s degree.
Assuming the population of the state aged 25 years or more is Bernoulli's population.
So, when 1 person aged 25 years or more from the state selected randomly, the probability of that person, p, having a bachelor’s degree,
[tex]p= 31/100=0.31\cdots(i)[/tex]
Now, according to Bernoulli's formula, the probability of exactly r success from the total number of sample n is
[tex]P(r)=\binom{n} {r}p^r(1-p)^{n-r}\cdots(ii)[/tex]
where p is the probability of success.
Here, a random sample of 50 residents of the state, age 25 years or older, will be selected.
So, n=50.
Given that variable B represents the number in the sample who have a bachelor’s degree,
We have to find the probability that B will equal 40.
So, r=B= 40.
Now, putting these values in equation(ii) and using p=0.25 from equation (i), we have
[tex]P(r=40)=\binom{50} {40}(0.31)^{40}(1-0.31)^{50-40}[/tex]
[tex]=\frac {50!}{40! (50-40)!}(0.31)^{40}(0.69)^{10} \\\\=\frac {50!}{40! \times 10!}(0.31)^{40}(0.69)^{10} \\\\=0.11\times 10^{-11}[/tex]
So, the probability of getting B=40 is [tex]0.11\times 10^{-11}[/tex] which is negligible.
Using the binomial distribution, it is found that there is a [tex]1.13 \times 10^{-12}[/tex] probability that B will equal 40.
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For each resident, there are only two possible outcomes. Either they have a bachelor's degree, or they do not. The probability of a resident having a bachelor degree is independent of any other resident, which means that the binomial distribution is used to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
p is the probability of a success on a single trial.
In this problem:
- 31% have a bachelor's degree, thus [tex]p = 0.31[/tex].
- Sample of 50, thus [tex]n = 50[/tex].
The probability is:
[tex]P(B = 40) = C_{50,40}(0.31)^{40}(0.69)^{10} = 1.13 \times 10^{-12}[/tex]
[tex]1.13 \times 10^{-12}[/tex] probability that B will equal 40.
A similar problem is given at https://brainly.com/question/24863377
