Answer:
The probability of getting B=40 is
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)](https://tex.z-dn.net/?f=p%3D%2031%2F100%3D0.31%5Ccdots%28i%29)
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)](https://tex.z-dn.net/?f=P%28r%29%3D%5Cbinom%7Bn%7D%20%7Br%7Dp%5Er%281-p%29%5E%7Bn-r%7D%5Ccdots%28ii%29)
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}](https://tex.z-dn.net/?f=P%28r%3D40%29%3D%5Cbinom%7B50%7D%20%7B40%7D%280.31%29%5E%7B40%7D%281-0.31%29%5E%7B50-40%7D)
![=\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}](https://tex.z-dn.net/?f=%3D%5Cfrac%20%7B50%21%7D%7B40%21%20%2850-40%29%21%7D%280.31%29%5E%7B40%7D%280.69%29%5E%7B10%7D%20%5C%5C%5C%5C%3D%5Cfrac%20%7B50%21%7D%7B40%21%20%5Ctimes%2010%21%7D%280.31%29%5E%7B40%7D%280.69%29%5E%7B10%7D%20%5C%5C%5C%5C%3D0.11%5Ctimes%2010%5E%7B-11%7D)
So, the probability of getting B=40 is
which is negligible.