Answer :
Answer:
D. 0.200, 0.026
Step-by-step explanation:
For each teen, there are only two possible outcomes. Either they have a fax machine, or they do not. The probability of a teen having a fax machine is independent of other teens. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
Forty-three percent of US teens have heard of a fax machine.
This means that [tex]p = 0.43[/tex]
You randomly select 12 US teens.
This means that [tex]n = 12[/tex]
Find the probability that the number of these selected teens that have heard of a fax machine is exactly six (first answer listed below).
This is [tex]P(X = 6)[/tex]. So
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 6) = C_{12,6}.(0.43)^{6}.(0.57)^{6} = 0.200[/tex]
So the answer is either option B or D.
Find the probability that the number is more than 8
[tex]P(X > 8) = P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12)[/tex]
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 9) = C_{12,9}.(0.43)^{9}.(0.57)^{3} = 0.021[/tex]
[tex]P(X = 10) = C_{12,10}.(0.43)^{10}.(0.57)^{2} = 0.005[/tex]
[tex]P(X = 11) = C_{12,11}.(0.43)^{11}.(0.57)^{1} = 0[/tex]
[tex]P(X = 12) = C_{12,12}.(0.43)^{12}.(0.57)^{0} = 0[/tex]
[tex]P(X > 8) = P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12) = 0.021 + 0.005 = 0.026[/tex]
So the correct answer is given by option D.