A poll is given, showing 70% are in favor of a new building project.
If 7 people are chosen at random, what is the probability that exactly 5 of them favor the new building project?
Well, if these people are randomly chosen, they better hope they're not randomly chosen by you! But let's do some math anyway.
The probability of randomly choosing a person who favors the new building project is 70%. So the probability of choosing 5 people who favor it out of 7 is:
(0.7)^5 * (0.3)^2
Raise your hand if you thought we would need to check your work! Oh right, I can't see you... Anyway, after doing the calculations, we find that the probability is approximately 0.1323, or about 13.23%. So it looks like there's a fair chance you'll find 5 supporters out of those 7 people. Good luck with your project!
To find the probability of exactly 5 out of 7 people favoring the new building project, we need to use the binomial probability formula:
P(X=k) = C(n,k) * p^k * (1-p)^(n-k)
Where:
P(X=k) is the probability of getting exactly k successes.
n is the total number of trials (number of people chosen).
k is the number of successful trials (number of people favoring the new building project).
p is the probability of success (proportion in favor of the project).
C(n,k) is the number of combinations of n items taken k at a time (n choose k).
In this case, n = 7, k = 5, and p = 0.70 (70% in favor).
Plugging the values into the formula:
P(X=5) = C(7,5) * 0.70^5 * (1-0.70)^(7-5)
C(7,5) = 7! / (5! * (7-5)!)
= 7! / (5! * 2!)
= (7 * 6 * 5!) / (5! * 2)
= 7 * 6 / 2
= 21
P(X=5) = 21 * 0.70^5 * (1-0.70)^(7-5)
= 21 * 0.16807 * 0.09
= 0.318087
Therefore, the probability that exactly 5 out of 7 people favor the new building project is approximately 0.318.
To determine the probability of exactly 5 out of 7 people favoring the new building project, we need to know the probability that a randomly chosen person favors the project. Let's assume the probability is p.
Since we are given that 70% of the respondents are in favor, we can conclude that p = 0.70.
To find the probability of exactly 5 people favoring the project, we apply the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k),
where n is the total number of trials, k is the number of "successes" (in this case, favoring the project), p is the probability of success, and "choose" denotes the binomial coefficient.
In our case, n = 7, k = 5, and p = 0.70. Plugging these values into the formula, we can calculate the desired probability:
P(X = 5) = (7 choose 5) * 0.70^5 * (1 - 0.70)^(7 - 5)
Calculating (7 choose 5) = 7! / (5! * (7-5)!) = 21, and substituting the values, we get:
P(X = 5) = 21 * 0.70^5 * 0.30^2
Calculating further, we find:
P(X = 5) ≈ 0.3087
Therefore, the probability of exactly 5 out of 7 randomly chosen people favoring the new building project is approximately 0.3087, or 30.87%.
Pr(exactly5)=.7^5 * .3^2 * C(7,5)
C(7,5)=7!/5!*2!=21
Pr= 21*.7^5*.3^2=.318