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Let's break down the information from the table and solve the questions:

The table represents the probability distribution of the number of cars owned by a college student:

| \( x \)  | \( P(x) \) |
|---------|-------------|
| 0       | 0.49        |
| 1       | 0.33        |
| 2       | 0.12        |
| 3       | 0.06        |

### Part (a): Is this a valid discrete probability distribution?

To check if this is a valid discrete probability distribution, we need to ensure:
1. All probabilities \( P(x) \) are between 0 and 1.
2. The sum of all probabilities \( P(x) \) must equal 1.

Let's verify:

\[
0.49 + 0.33 + 0.12 + 0.06 = 1.00
\]

Since the sum equals 1 and all probabilities are between 0 and 1, **yes, this is a valid discrete probability distribution**.

### Part (b): Find the mean number of cars owned.

The mean (or expected value) of a discrete probability distribution is calculated using the formula:

\[
\mu = \sum [x \cdot P(x)]
\]

Let's compute it:

\[
\mu = (0 \times 0.49) + (1 \times 0.33) + (2 \times 0.12) + (3 \times 0.06)
\]

Breaking it down:

\[
\mu = 0 + 0.33 + 0.24 + 0.18 = 0.75
\]

So, the mean number of cars owned is **0.75**.

---

Do you want further details or have any questions? Here are some related questions to deepen understanding:

1. How do we calculate the variance of a probability distribution?
2. What is the interpretation of the expected value in this context?
3. What would be the probability of owning more than 1 car according to this distribution?
4. How would the distribution change if the probability of owning 0 cars increased?
5. How do we compute the standard deviation from the given probability distribution?

**Tip:** The expected value represents the long-term average or mean outcome of a random variable over many trials.

The following table represents the probability of the number of cars owned by a college student. Is this a valid discrete probability distribution? Find the mean number of cars owned.

This problem explores the probability distribution of cars owned by college students. It verifies if the distribution is valid and calculates the mean (expected value). Suitable for students in Grades 10-12, this example covers core probability and statistics concepts.

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