Answer :
Answer: (D) 16%
Step-by-step explanation:
Binomial probability formula :-
[tex]P(x)=^nC_xp^x(1-p)^n-x[/tex], where n is the sample size , p is population proportion and P(x) is the probability of getting success in x trial.
Given : The proportion of students in College are near-sighted : p= 0.28
Sample size : n= 20
Then, the the probability that in a randomly chosen group of 20 College students, exactly 4 are near-sighted is given by :_
[tex]P(x=4)=^{20}C_4(0.28)^4(1-0.28)^{20-4}\\\\=\dfrac{20!}{4!16!}(0.28)^4(0.72)^{16}\\\\=0.155326604912\approx0.16\%[/tex]
Hence, the probability that in a randomly chosen group of 20 College students, exactly 4 are near-sighted is closest to 16%.
We will see that the probability is closest to 16%, so the correct option is D.
How to get the probability?
We know that 28% of the students are near-sighted, then the other 72% arent.
This means that:
- There is a probability of 0.28 that a random student is near-sighted.
- There is a probability of 0.72 that a random student is not near-sighted.
Then the probability that out of 20, there are 4 near-sighted students is:
[tex]P = (0.72)^{16}*(0.28)^4*C(20, 4)[/tex]
Where C(20, 4) is the number of different possible groups of 4 that we can make out from the 20 students, it is given by:
[tex]C(N, K) = \frac{N!}{(N - K)!*K!}[/tex]
Then we have:
[tex]P = (0.72)^{16}*(0.28)^4*\frac{20!}{16!*4!}\\ \\P = (0.72)^{16}*(0.28)^4*\frac{20*19*18*17}{4*3*2} = 0.155[/tex]
Written in percentage form, and rounding to the nearest whole number, we have:
P = (0.155)*100% = 15.5% ≈ 16%
So the correct option is D.
If you want to learn more about probability, you can read:
https://brainly.com/question/251701