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To determine if the given table represents a probability distribution, we need to check two key conditions for probability distributions:

1. **Non-negativity:** All probabilities \( P(x) \) should be non-negative, meaning \( P(x) \geq 0 \) for all \( x \).
2. **Sum of probabilities equals 1:** The total sum of all probabilities should equal 1, i.e., \( \sum P(x) = 1 \).

### Table:

| \( x \) | \( P(x) \) |
|--------|------------|
| 0      | 0.3        |
| 1      | 0.2        |
| 2      | 0.15       |
| 3      | 0.3        |

#### Step 1: Check Non-Negativity
All the probabilities in the table are non-negative: \( 0.3, 0.2, 0.15, 0.3 \). So, the first condition is satisfied.

#### Step 2: Sum of Probabilities
Now let's add up all the probabilities:
\[
0.3 + 0.2 + 0.15 + 0.3 = 0.95
\]
The sum of the probabilities is 0.95, which is **not equal to 1**.

#### Conclusion:
Since the sum of probabilities does not equal 1, this is **not a probability distribution**.

Would you like further details or have any questions?

### 5 related questions:
1. What is the correct condition for a valid probability distribution?
2. How do you check for non-negativity in probability tables?
3. What are some real-world examples of probability distributions?
4. What happens if the sum of probabilities is more than 1?
5. How can we adjust this table to make it a valid probability distribution?

**Tip:** Always double-check the sum of probabilities when verifying probability distributions!

Determine whether this table represents a probability distribution.

Explore whether a given table with probabilities for events represents a valid probability distribution. Understand key concepts like non-negativity and the requirement that the sum of probabilities equals 1. This guide is suitable for high school students in Grades 10-12.

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