Answer:
a) 0.2076
b) 0.3216
c) 0.114
Explanation:
We are given the following information:
We treat adult on a diet as a success.
P(Adult on diet) = [tex]\frac{1}{4}[/tex] = 0.25
Then the number of adults follows a binomial distribution, where
[tex]P(X=x) = \binom{n}{x}.p^x.(1-p)^{n-x}[/tex]
where n is the total number of observations, x is the number of success, p is the probability of success.
Now, we are given n = 8
We have to evaluate:
(a) exactly three
[tex]P(x =3)\\\\P(x =3= \binom{8}{3}(0.25)^{3}(1-0.25)^{5} \\\\P(x =3)=0.2076[/tex]
(b) at least three
[tex]P(x \geq 3) = 1 -P(x = 0) - P(x = 1) - P(x = 2)\\\\=1- \binom{8}{0}(0.25)^0(1-0.25)^8-\binom{8}{1}(0.25)^1(1-0.25)^7 -\binom{8}{2}(0.25)^2(1-0.25)^6\\=1-0.1001-0.2669-0.3114\\= 0.3216[/tex]
(c) more than three
[tex]P(x > 3) = 1 -P(x = 0) - P(x = 1) - P(x = 2) - P(x=3)\\\\=1- \binom{8}{0}(0.25)^0(1-0.25)^8-\binom{8}{1}(0.25)^1(1-0.25)^7 -\binom{8}{2}(0.25)^2(1-0.25)^6\\ -\binom{8}{3}(0.25)^3(1-0.25)^5 \\=1-0.1001-0.2669-0.3114-0.2076\\= 0.114[/tex]
