Let's recheck the calculations to confirm the expected value. 

### Given Data:
- **Number of New Clients (x):** 0, 1, 2, 3, 4, 5, 6
- **Probability (P(x)):** 0.05, 0.10, 0.20, 0.30, 0.20, 0.10, 0.05

### Expected Value Calculation:
The expected value \( E(X) \) is calculated using the formula:
\[
E(X) = \sum_{i} x_i \cdot P(x_i)
\]
Let's plug in the values:

\[
E(X) = (0 \cdot 0.05) + (1 \cdot 0.10) + (2 \cdot 0.20) + (3 \cdot 0.30) + (4 \cdot 0.20) + (5 \cdot 0.10) + (6 \cdot 0.05)
\]

\[
E(X) = 0 + 0.10 + 0.40 + 0.90 + 0.80 + 0.50 + 0.30
\]

\[
E(X) = 3.1
\]

### Final Answer:
You're correct. The expected number of new clients per month is **3.1**.

Would you like further explanation or have any other questions?

### Related Questions:
1. How does rounding affect the interpretation of the expected value?
2. What are the implications of an expected value slightly above 3?
3. How would you calculate the variance for this distribution?
4. What is the probability of getting fewer than 3 clients?
5. How can this calculation help in resource planning?
6. What if the distribution were skewed? How would that impact the expected value?
7. How does the expected value change if the probabilities were adjusted?
8. How can we calculate the median of this distribution?

### Tip:
Always double-check your calculations to ensure accuracy, especially in statistical problems where small errors can lead to significant differences in interpretation.

Learn how to calculate the expected number of new clients per month using probability distribution. This problem involves determining the expected value based on different probabilities of new client acquisition. Suitable for advanced high school students and anyone interested in probability calculations.

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