Question

Provide an appropriate response.

In a recent survey, 72% of the community favored building a health center in their neighborhood. If 14 citizens
are chosen, find the probability that exactly 10 of them favor the building of the health center.
0.001
0.714
0.720
0.230

Answer 1

0.230

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;

`(p + q)^n = ^nC_0 .\ p^n.\ q^(0) + ....+ ^nC_r .\ p^r.\ q^(n-r)+ .. +^nC_n .\ p^0.\ q^(n)`

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

`n = 14`

`r = 10`

`p = 72\% = 0.72`

`q = 1 - p`

`q = 1 - 0.72`

`q = 0.28`

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

`Probability = ^nC_r .\ p^r.\ q^(n-r)`

`Probability = ^(14)C_(10) .\ 0.72^(10).\ 0.28^(14-10)`

`^(14)C_(10) = 1001`

So, the expression becomes

`Probability =1001 * \ 0.72^(10).\ 0.28^(14-10)`

`Probability =1001 * \ 0.72^(10).\ 0.28^4`

`Probability =1001 * 0.03743906242 * 0.00614656`

`Probability =0.23035156495`

`Probability =0.230` ----Approximated

Hence, the probability that exact;y 10 will favor the building of the health center is 0.230