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 \(\boxed{0.232}\).