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To determine whether this table represents a discrete probability distribution, we need to verify that it satisfies two key conditions for probability distributions:

1. **Non-negative Probabilities**: All values in the probability column, \( P(x) \), must be non-negative (i.e., each probability must be greater than or equal to zero).
2. **Sum of Probabilities Equals 1**: The probabilities of all outcomes must sum up to 1.

Let’s examine each condition for the given table:

### Given Table
| \( x \) | \( P(x) \) |
|---------|------------|
| 56      | 0.3        |
| 65      | 0.3        |
| 76      | 0.2        |
| 86      | -0.3       |

1. **Non-negative Probabilities**: The value \( P(86) = -0.3 \) is negative, which violates the first condition.
2. **Sum of Probabilities**: If we were to add up the probabilities:
   \[
   0.3 + 0.3 + 0.2 + (-0.3) = 0.5
   \]
   This sum is not equal to 1, which violates the second condition.

Since both conditions are violated, this table **does not represent a discrete probability distribution**.

Would you like further details or have any other questions about this solution?

### Related Questions
1. What are the basic requirements for a probability distribution?
2. How would you adjust the values in this table to make it a valid probability distribution?
3. What distinguishes a discrete probability distribution from a continuous one?
4. Can probabilities ever be negative in any kind of probability model?
5. How do you calculate the mean of a discrete probability distribution?

### Tip
When verifying a probability distribution, always check both the sum of probabilities and ensure all probabilities are non-negative.

Determine whether the table represents a discrete probability distribution. Explain why or why not.

Learn how to determine if a given table represents a discrete probability distribution. The solution explains the key conditions: non-negative probabilities and a total probability sum of 1. This example includes a table with values that violate these conditions, making it an ideal learning resource for college-level statistics.

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