Explanation:
To find the probability that exactly 1 out of 3 people chosen at random favor the new building project when 80% are in favor, you can use the binomial probability formula.
The formula for the probability of exactly k successes in n trials, where the probability of success on a single trial is p, is:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
In this case:
- n (number of trials) is 3.
- k (number of successes) is 1.
- p (probability of success on a single trial) is 0.80 (since 80% are in favor).
Using the formula, you can calculate the probability:
P(X = 1) = (3 choose 1) * (0.80)^1 * (1 - 0.80)^(3 - 1)
P(X = 1) = (3 choose 1) * 0.80 * 0.20^2
You can calculate "3 choose 1" as 3 (it represents the number of ways to choose 1 success out of 3 trials). Now, plug these values into the formula:
P(X = 1) = 3 * 0.80 * 0.20^2
P(X = 1) = 3 * 0.80 * 0.04
P(X = 1) = 0.096
So, the probability that exactly 1 out of 3 people chosen at random favor the new building project is 0.096 or 9.6%.