34% 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 0.34.

To construct a binomial distribution with \( n = 8 \) and \( p = 0.34 \), we will calculate the probability for each possible number of employees (from 0 to 8) who judge their peers by the cleanliness of their workspaces.

The binomial probability formula is given by:

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

where:
- \( n \) is the number of trials (employees surveyed), \( n = 8 \).
- \( k \) is the number of successful trials (employees who judge their peers), ranging from 0 to 8.
- \( p \) is the probability of success on each trial, \( p = 0.34 \).
- \(\binom{n}{k}\) is the binomial coefficient, calculated as \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

Let's calculate the probabilities for \( k = 0 \) to \( k = 8 \):

1. **For \( k = 0 \)**:
\[
P(X = 0) = \binom{8}{0} (0.34)^0 (0.66)^8 = 1 \cdot 1 \cdot (0.66)^8 \approx 0.0214
\]

2. **For \( k = 1 \)**:
\[
P(X = 1) = \binom{8}{1} (0.34)^1 (0.66)^7 = 8 \cdot 0.34 \cdot (0.66)^7 \approx 0.0881
\]

3. **For \( k = 2 \)**:
\[
P(X = 2) = \binom{8}{2} (0.34)^2 (0.66)^6 = 28 \cdot (0.34)^2 \cdot (0.66)^6 \approx 0.1853
\]

4. **For \( k = 3 \)**:
\[
P(X = 3) = \binom{8}{3} (0.34)^3 (0.66)^5 = 56 \cdot (0.34)^3 \cdot (0.66)^5 \approx 0.2612
\]

5. **For \( k = 4 \)**:
\[
P(X = 4) = \binom{8}{4} (0.34)^4 (0.66)^4 = 70 \cdot (0.34)^4 \cdot (0.66)^4 \approx 0.2274
\]

6. **For \( k = 5 \)**:
\[
P(X = 5) = \binom{8}{5} (0.34)^5 (0.66)^3 = 56 \cdot (0.34)^5 \cdot (0.66)^3 \approx 0.1233
\]

7. **For \( k = 6 \)**:
\[
P(X = 6) = \binom{8}{6} (0.34)^6 (0.66)^2 = 28 \cdot (0.34)^6 \cdot (0.66)^2 \approx 0.0419
\]

8. **For \( k = 7 \)**:
\[
P(X = 7) = \binom{8}{7} (0.34)^7 (0.66)^1 = 8 \cdot (0.34)^7 \cdot 0.66 \approx 0.0080
\]

9. **For \( k = 8 \)**:
\[
P(X = 8) = \binom{8}{8} (0.34)^8 (0.66)^0 = 1 \cdot (0.34)^8 \cdot 1 \approx 0.0005
\]

Thus, the binomial distribution for \( n = 8 \) and \( p = 0.34 \) is as follows:

\[
\begin{array}{c|c}
k & P(X = k) \\
\hline
0 & 0.0214 \\
1 & 0.0881 \\
2 & 0.1853 \\
3 & 0.2612 \\
4 & 0.2274 \\
5 & 0.1233 \\
6 & 0.0419 \\
7 & 0.0080 \\
8 & 0.0005 \\
\end{array}
\]

Would you like more details on how these calculations were made, or do you have any other questions?

Here are 8 related questions to further explore the topic:

1. What is the mean of this binomial distribution?
2. What is the variance of this binomial distribution?
3. How do you calculate the cumulative probability for \( P(X \leq 3) \)?
4. What is the probability that exactly 5 employees judge their peers by cleanliness?
5. How does changing the probability \( p \) to 0.5 affect the distribution?
6. What are the skewness and kurtosis of this binomial distribution?
7. Can you use the normal approximation for this binomial distribution?
8. How does increasing the number of employees surveyed to 20 affect the probabilities?

**Tip:** Understanding the binomial distribution helps in solving problems where the outcomes of interest occur in independent trials with a constant probability of success.

Explore how to construct a binomial distribution with 8 employees and a probability of 0.34 for judging peers based on workspace cleanliness. Calculate probabilities for each possible outcome from 0 to 8 employees judging their peers.

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