38​% of employees judge their peers by the cleanliness of their workspaces. You randomly select 8 employees and ask them whether they judge their peers by the cleanliness of their workspaces. The random variable represents the number of employees who judge their peers by the cleanliness of their workspaces.

Construct a binomial distribution using n equals 8 and p equals 0.38.

To construct a binomial distribution with \( n = 8 \) (the number of trials) and \( p = 0.38 \) (the probability of success on each trial), we can calculate the probability for each possible value of the random variable \( X \), which represents the number of employees who judge their peers by the cleanliness of their workspaces. 

The binomial probability formula is:

\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]

where:
- \( n = 8 \) is the number of trials,
- \( k \) is the number of successes (which ranges from 0 to 8),
- \( p = 0.38 \) is the probability of success on each trial,
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \).

### Calculating the Probabilities for Each \( k \):

Let's calculate \( P(X = k) \) for \( k = 0 \) through \( k = 8 \).

\[
\text{For } k = 0: P(X = 0) = \binom{8}{0} (0.38)^0 (0.62)^8
\]
\[
\text{For } k = 1: P(X = 1) = \binom{8}{1} (0.38)^1 (0.62)^7
\]
\[
\text{For } k = 2: P(X = 2) = \binom{8}{2} (0.38)^2 (0.62)^6
\]
\[
\text{For } k = 3: P(X = 3) = \binom{8}{3} (0.38)^3 (0.62)^5
\]
\[
\text{For } k = 4: P(X = 4) = \binom{8}{4} (0.38)^4 (0.62)^4
\]
\[
\text{For } k = 5: P(X = 5) = \binom{8}{5} (0.38)^5 (0.62)^3
\]
\[
\text{For } k = 6: P(X = 6) = \binom{8}{6} (0.38)^6 (0.62)^2
\]
\[
\text{For } k = 7: P(X = 7) = \binom{8}{7} (0.38)^7 (0.62)^1
\]
\[
\text{For } k = 8: P(X = 8) = \binom{8}{8} (0.38)^8 (0.62)^0
\]

I'll calculate these values next.Here is the binomial distribution for \( n = 8 \) and \( p = 0.38 \):

\[
\begin{array}{|c|c|}
\hline
\textbf{k} & \textbf{P(X = k)} \\
\hline
0 & 0.0218 \\
1 & 0.1071 \\
2 & 0.2297 \\
3 & 0.2815 \\
4 & 0.2157 \\
5 & 0.1058 \\
6 & 0.0324 \\
7 & 0.0057 \\
8 & 0.0004 \\
\hline
\end{array}
\]

These probabilities represent the likelihood of \( k \) employees (out of 8) judging their peers by the cleanliness of their workspaces.

Would you like more details or have any questions?

Here are some related questions you might explore:
1. What is the expected value of the random variable \( X \)?
2. How do you calculate the cumulative probability for at most 3 employees judging their peers?
3. What is the probability that exactly 4 employees judge their peers?
4. How does the probability distribution change if \( p \) were 0.5?
5. How do you interpret the skewness of this distribution?

**Tip:** The sum of all probabilities in a binomial distribution should always equal 1, which helps to check your calculations.

Learn how to construct a binomial distribution with n = 8 and p = 0.38. Understand the probabilities for the number of employees judging their peers by the cleanliness of their workspaces. Detailed calculations and step-by-step explanations provided.

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