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To solve part (a) of the question, we need to construct a probability distribution using the provided frequency distribution. Here's the step-by-step process:

### Step 1: Calculate the Total Number of Households
- The frequency distribution provides the number of households with different numbers of televisions.
- Add up the number of households in each category:

\[
29 + 447 + 721 + 1404
\]

### Step 2: Calculate the Probability \( P(x) \)
- The probability \( P(x) \) for a given number of televisions \( x \) is defined as:

\[
P(x) = \frac{\text{Number of households with } x \text{ televisions}}{\text{Total number of households}}
\]

### Applying these steps, I'll compute the probabilities for \( x = 0, 1, 2, 3 \).### Probability Distribution
The total number of households is 2,601. The probability distribution for the number of televisions per household is as follows:

- \( P(0) = \frac{29}{2601} \approx 0.011 \)
- \( P(1) = \frac{447}{2601} \approx 0.172 \)
- \( P(2) = \frac{721}{2601} \approx 0.277 \)
- \( P(3) = \frac{1404}{2601} \approx 0.540 \)

These values rounded to three decimal places are:
- **\( P(0) = 0.011 \)**
- **\( P(1) = 0.172 \)**
- **\( P(2) = 0.277 \)**
- **\( P(3) = 0.540 \)**

Would you like me to assist with part (b) or any further questions?

### Related Questions:
1. How do you verify that the probabilities form a valid distribution?
2. How would you interpret \( P(x) = 0.540 \) for \( x = 3 \)?
3. What would the histogram for this distribution look like?
4. What is the expected value of the number of televisions?
5. How would the distribution change if a new category with 4 televisions is added?

### Tip:
When constructing a probability distribution, always ensure that the sum of all probabilities equals 1.

Use the frequency distribution of the number of televisions per household in a small town to construct a probability distribution.

Learn how to construct a probability distribution using a frequency distribution of the number of televisions per household in a small town. This problem involves calculating probabilities based on frequency data. Suitable for students in Grades 10-12.

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