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### Given Table:
| Household Size (x) | 1     | 2     | 3     | 4     | 5     | 6     | 7     |
|--------------------|-------|-------|-------|-------|-------|-------|-------|
| p(x)               | 0.217 | 0.363 | 0.165 | 0.145 | 0.026 | 0.018 | ?     |

---

### a. **Complete the probabilities for the table.**
To complete the table, the probabilities must sum to 1. First, we sum the provided probabilities:

\[
0.217 + 0.363 + 0.165 + 0.145 + 0.026 + 0.018 = 0.934
\]

Thus, the missing probability for household size 7 (p(7)) is:

\[
p(7) = 1 - 0.934 = 0.066
\]

So, the completed probability for \( x = 7 \) is 0.066.

---

### b. **What is the probability of exactly 1 for the household size?**
From the table, the probability for household size 1 is given as:

\[
p(1) = 0.217
\]

---

### c. **What is the probability that there are 1 or two people in the household?**
This is the sum of the probabilities for household sizes 1 and 2:

\[
P(\text{1 or 2 people}) = p(1) + p(2) = 0.217 + 0.363 = 0.580
\]

---

### d. **What is the probability of at least 5 for the household size?**
This is the sum of the probabilities for household sizes 5, 6, and 7:

\[
P(\text{at least 5 people}) = p(5) + p(6) + p(7) = 0.026 + 0.018 + 0.066 = 0.110
\]

---

### e. **What is the probability of less than 3 for the household size?**
This is the sum of the probabilities for household sizes 1 and 2:

\[
P(\text{less than 3 people}) = p(1) + p(2) = 0.217 + 0.363 = 0.580
\]

---

### f. **What is the expected number of people for a household size?**
The expected value \( E(X) \) of a probability distribution is calculated as:

\[
E(X) = \sum (x \cdot p(x))
\]

Let's calculate it using the provided probabilities:

\[
E(X) = (1 \cdot 0.217) + (2 \cdot 0.363) + (3 \cdot 0.165) + (4 \cdot 0.145) + (5 \cdot 0.026) + (6 \cdot 0.018) + (7 \cdot 0.066)
\]

\[
E(X) = 0.217 + 0.726 + 0.495 + 0.580 + 0.130 + 0.108 + 0.462 = 2.718
\]

Thus, the expected number of people per household is approximately **2.72**.

---

Would you like further clarification on any part of the solution? 

Here are 5 related questions to expand on this problem:
1. How is variance calculated for this probability distribution?
2. What is the cumulative probability for a household size less than 5?
3. How would the probabilities change if the distribution were altered by a new census?
4. What does the probability distribution tell us about the most common household size?
5. How could the expected household size be interpreted in real-life scenarios?

**Tip:** When dealing with probability distributions, always verify that the sum of the probabilities equals 1 to ensure the distribution is valid.

Complete the probabilities and solve related probability questions for household size data.

Learn how to solve probability-related questions based on household size data. This includes completing a probability table, finding specific probabilities, calculating cumulative probabilities, and determining the expected number of people in a household.

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