Assume that random guesses are made for 8 multiple-choice questions on a test with 5 choices for each question, so that there are $n=8$ trials, each with probability of success (correct) given by $p=0.20$. Find the probability of no correct answers. Click on the icon to view the binomial probability table. The probability of no correct answers is (Round to three decimal places as needed.)

Step 1 ：We are given a problem where random guesses are made for 8 multiple-choice questions on a test with 5 choices for each question. We are asked to find the probability of no correct answers.

Step 2 ：This is a binomial distribution problem. The binomial distribution model deals with finding the probability of success of an event which has only two possible outcomes in a series of experiments.

Step 3 ：For a random variable X if X is B(n, p) where n is the number of trials, and p is the probability of success on each trial, we can find the probability of getting exactly k successes among n trials is given by the function: \(P(X=k) = C(n, k) * (p^k) * (1-p)^(n-k)\) where C(n, k) is the binomial coefficient.

Step 4 ：In this case, we are looking for the probability of no correct answers, which means k=0. So we can substitute n=8, p=0.20 and k=0 into the formula and calculate the result.

Step 5 ：Substituting the given values into the formula, we get \(P(X=0) = C(8, 0) * (0.2^0) * (1-0.2)^(8-0)\)

Step 6 ：Solving the above expression, we get \(P = 0.1677721600000001\)

Step 7 ：Rounding to three decimal places as needed, we get \(P = 0.168\)

Step 8 ：Final Answer: The probability of no correct answers is \(\boxed{0.168}\)