Car inspection: of all the registered automobiles in a city, $12 \%$ fail the emissions test. Fifteen automobiles are selected at random to undergo an emission test. Round the answers to at least four decimal places.
Part: 0 / 4
Part 1 of 4
(a) Find the probability that exactly four of them fail the test.
The probability that exactly four of them fail the test is

Step 1 ：This problem is about binomial distribution. The probability of success (failing the test) is 0.12, the number of trials is 15, and we want to find the probability of exactly 4 successes.

Step 2 ：The formula for the binomial probability is: \(P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))\), where \(P(X=k)\) is the probability of \(k\) successes, \(C(n, k)\) is the combination of \(n\) items taken \(k\) at a time, \(p\) is the probability of success, \(n\) is the number of trials, and \(k\) is the number of successes.

Step 3 ：Let's plug in the values into the formula: \(p = 0.12\), \(n = 15\), and \(k = 4\).

Step 4 ：The combination of 15 items taken 4 at a time is 1365.

Step 5 ：Calculate the probability: \(prob = 0.06936925481723392\).

Step 6 ：Round the answer to at least four decimal places, the final answer is: \(\boxed{0.0694}\).