Answer:
0.230
Step-by-step explanation:
Given
Estimate = 72%
Number of citizens = 14
Required
Find the probability that exactly 10 of the citizens will be in favor
This question can be solved using binomial expansion of probability which states;
[tex](p + q)^n = ^nC_0 .\ p^n.\ q^{0} + ....+ ^nC_r .\ p^r.\ q^{n-r}+ .. +^nC_n .\ p^0.\ q^{n}[/tex]
Where p and q are the probabilities of those in favor and against of building a health center;
n is the selected sample and r is the sample in favor
So; from the above analysis
[tex]n = 14[/tex]
[tex]r = 10[/tex]
[tex]p = 72\% = 0.72[/tex]
[tex]q = 1 - p[/tex]
[tex]q = 1 - 0.72[/tex]
[tex]q = 0.28[/tex]
Since, we're solving for the probability that exactly 10 citizens will be in favor;
we'll make use of
Substituting these values in the formula above
[tex]Probability = ^nC_r .\ p^r.\ q^{n-r}[/tex]
[tex]Probability = ^{14}C_{10} .\ 0.72^{10}.\ 0.28^{14-10}[/tex]
[tex]^{14}C_{10} = 1001[/tex]
So, the expression becomes
[tex]Probability =1001 * \ 0.72^{10}.\ 0.28^{14-10}[/tex]
[tex]Probability =1001 * \ 0.72^{10}.\ 0.28^4[/tex]
[tex]Probability =1001 * 0.03743906242 * 0.00614656[/tex]
[tex]Probability =0.23035156495[/tex]
[tex]Probability =0.230[/tex] ----Approximated
Hence, the probability that exact;y 10 will favor the building of the health center is 0.230
