Determine the probability $P(6)$ for a binomial experiment with $n=14$ trials and success probability $p=0.2$. Then find the mean, variance, and standard deviation. Part 1 of 3 Determine the probability $P(6)$. Round the answer to at least four decimal places. \[ P(6)=0.0322 \] Part: 1 / 3 Part 2 of 3 Find the mean. Round the answer to two decimal places. The mean is \[ \times 5 \] Skip Part Check

Step 1 ：The first part of the question asks for the probability of getting 6 successes in a binomial experiment with 14 trials and a success probability of 0.2. The formula for the probability mass function of a binomial distribution is: \[P(k; n, p) = \binom{n}{k} p^k (1-p)^{n-k}\] where: \(\binom{n}{k}\) is the number of combinations of n items taken k at a time, \(p\) is the probability of success, \(k\) is the number of successes, \(n\) is the number of trials.

Step 2 ：We can plug in the given values into this formula to find the probability. Let's denote: n = 14, k = 6, p = 0.2.

Step 3 ：The calculated probability is approximately 0.0322.

Step 4 ：Final Answer: The probability $P(6)$ for a binomial experiment with $n=14$ trials and success probability $p=0.2$ is approximately \(\boxed{0.0322}\).