Final answer:
The probability that exactly 4 out of 9 people are in favor of a new building project, given that 55% are in favor, is approximately 20.23%.
Step-by-step explanation:
The subject of this question is Mathematics, and the grade level is most likely High School as it deals with probability and combinatorics which are typically covered at that level.
To find the probability that exactly 4 out of 9 people chosen at random are in favor of a new building project, we can use the binomial probability formula:
P(X=k) = (n C k) * p^k * (1-p)^(n-k)
Where:
- n is the total number of trials (in this case, n = 9),
- k is the number of successful trials we want to find the probability for (k = 4),
- p is the probability of success on a single trial (p = 0.55 as 55% are in favor), and
- (n C k) is the binomial coefficient.
First, calculate the binomial coefficient (9 C 4), which is the number of combinations of 9 items taken 4 at a time:
(9 C 4) = 9! / (4! * (9-4)!) = 126
Then, use the probability value for p and (1-p) to calculate the probability that exactly 4 out of 9 people favor the project:
P(X=4) = (9 C 4) * 0.55^4 * 0.45^5
P(X=4) = 126 * 0.09150625 * 0.018494125
P(X=4) ≈ 0.202331
So, the probability that exactly 4 of the 9 randomly chosen people are in favor of the new building project is approximately 20.23%.