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
\[
\times 5
\]
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Answer
Final Answer: The probability $P(6)$ for a binomial experiment with $n=14$ trials and success probability $p=0.2$ is approximately \(\boxed{0.0322}\).
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 \(\boxed{0.0322}\).
