Final answer:
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.