According to a recent survey, $89 \%$ of students say that they do not get enough sleep. Assume the survey meets the conditions of a binomial experiment. Round your answers to three decimal places

In a random survey of 30 students, find the probability that exactly 27 of them will say that they do not get enough sleep:

Step 1 ：The problem is asking for the probability of a specific outcome in a binomial experiment. The binomial distribution formula can be used to solve this problem. The formula is: \(P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))\) where: \(P(X=k)\) is the probability of k successes in n trials, \(C(n, k)\) is the combination of n items taken k at a time, p is the probability of success on a single trial, n is the number of trials, and k is the number of successes.

Step 2 ：In this case, n=30 (the number of students surveyed), k=27 (the number of students who say they do not get enough sleep), and p=0.89 (the probability that a student says they do not get enough sleep).

Step 3 ：Substitute the given values into the formula: \(P(X=27) = C(30, 27) * (0.89^{27}) * ((1-0.89)^{30-27})\)

Step 4 ：Calculate the combination: \(C(30, 27) = 4060\)

Step 5 ：Calculate the probability: \(P(X=27) = 4060 * (0.89^{27}) * ((1-0.89)^{30-27}) = 0.232\)

Step 6 ：Final Answer: The probability that exactly 27 out of 30 students will say that they do not get enough sleep is \(\boxed{0.232}\).