Final answer:
Using the binomial probability formula, the probability that exactly 4 out of the 10 residents recognize the brand name is approximately 0.0604. So the correct option is d.
Step-by-step explanation:
To find the probability that exactly 4 out of the 10 Coffleton residents recognize the brand name, when the recognition rate is 54%, we can 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 k successes in n trials
- C(n, k) is the binomial coefficient (combinations of n items taken k at a time)
- p is the probability of success on a single trial
- n is the number of trials
- k is the number of successes
Plugging the values into the formula we get:
P(X = 4) = C(10, 4) * 0.54^4 * (1 - 0.54)^(10 - 4)
P(X = 4) = (10! / (4!(10 - 4)!)) * 0.54^4 * 0.46^6
P(X = 4) = 210 * 0.54^4 * 0.46^6
P(X = 4) ≈ 0.0604
Therefore, the probability that exactly 4 of the 10 residents recognize the brand name is approximately 0.0604, which corresponds to option d.