Question

According to a recent survey, 31 percent of the residents of a certain state who are age 25 years or older have a bachelor’s degree. A random sample of 50 residents of the state, age 25 years or older, will be selected. Let the random variable B represent the number in the sample who have a bachelor’s degree. What is the probability that B will equal 40 ?

Answer 1

Answer:

(50 (0.31)^40(0.69)^10

40

Answer 2

Answer:

The probability of  getting B=40 is $0.11\times 10^{-11}$ 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,

$p= 31/100=0.31\cdots(i)$

Now, according to Bernoulli's formula, the probability of exactly r success from the total number of sample n is

$P(r)=\binom{n} {r}p^r(1-p)^{n-r}\cdots(ii)$

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

$P(r=40)=\binom{50} {40}(0.31)^{40}(1-0.31)^{50-40}$

$=\frac {50!}{40! (50-40)!}(0.31)^{40}(0.69)^{10} \\\\=\frac {50!}{40! \times 10!}(0.31)^{40}(0.69)^{10} \\\\=0.11\times 10^{-11}$

So, the probability of  getting B=40 is $0.11\times 10^{-11}$ which is negligible.