A binomial experiment with probability of success \( p=0.85 \) and \( n=6 \) trials is conducted. What is the probability that the experiment results in exactly 5 suocesses? Do not round your intermediate computations, and round your answer to three decimal places. (If necessary, consult a list of formulas.)

To find the probability of exactly 5 successes in a binomial experiment with a probability of success \( p = 0.85 \) and \( n = 6 \) trials, we can use the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where: - \( P(X = k) \) is the probability of exactly \( k \) successes, - \( n \) is the number of trials, - \( k \) is the number of successes, - \( p \) is the probability of success on a single trial. Given: - \( p = 0.85 \) - \( n = 6 \) - \( k = 5 \) Substitute the values into the formula: \[ P(X = 5) = \binom{6}{5} (0.85)^5 (1-0.85)^{6-5} \] \[ P(X = 5) = \binom{6}{5} (0.85)^5 (0.15)^1 \] \[ P(X = 5) = 6 \times (0.85)^5 \times 0.15 \] Now, we can calculate the probability using the formula. Calculate the value by following steps: - step0: Calculate: \(6\times 0.85^{5}\times 0.15\) - step1: Convert the expressions: \(6\left(\frac{17}{20}\right)^{5}\times 0.15\) - step2: Multiply the terms: \(0.9\left(\frac{17}{20}\right)^{5}\) - step3: Convert the expressions: \(\frac{9}{10}\left(\frac{17}{20}\right)^{5}\) - step4: Multiply the numbers: \(\frac{9\times 17^{5}}{10\times 20^{5}}\) The probability of exactly 5 successes in the binomial experiment with a probability of success \( p = 0.85 \) and \( n = 6 \) trials is approximately 0.399335.