Question

Provide an appropriate response.

In a recent survey, 81% of the community favored building a health center in their neighborhood. If 14 citizens are chosen, find the probability that exactly 6 of them favor the building of the health center.
a. 0.026
b. 0.810
c. 0.001
d. 0.429

Answer 1

Using the binomial probability formula, the probability that exactly 6 out of 14 citizens favor the building of a health center is approximately 0.026.

Step-by-step explanation:

To find the probability that exactly 6 out of 14 citizens favor the building of a health center, given that 81% of the community is in favor, we can use the binomial probability formula:

P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)

Where:

X = number of successes

C(n, k) = combination of n items taken k at a time

p = probability of success on a single trial

(1 - p) = probability of failure on a single trial

Plugging in the values, we get:

P(X = 6) = C(14, 6) * 0.81^6 * (1 - 0.81)^(14 - 6)

Calculating further:

C(14, 6) = 14! / (6! * (14 - 6)!) = 3003

P(X = 6) = 3003 * 0.81^6 * 0.19^8

After calculating the numerical value, we get a probability of approximately 0.026.

Therefore, the correct answer is a. 0.026.