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

1. **Non-negative probabilities**: Each probability \( P(X) \) must be between 0 and 1, inclusive.
2. **Sum of probabilities equals 1**: The sum of all probabilities \( P(X) \) should equal 1.

From the table:

\[
\begin{array}{|c|c|}
\hline
X & P(X) \\
\hline
7 & 0.1 \\
3 & 0.25 \\
1 & 0.15 \\
8 & 0.5 \\
\hline
\end{array}
\]

### Step 1: Check Non-Negative Probabilities
All probabilities \( P(X) \) are non-negative:
\[
0.1, 0.25, 0.15, 0.5 \geq 0
\]
So, this condition is satisfied.

### Step 2: Check Sum of Probabilities
Now, let's sum the probabilities:
\[
0.1 + 0.25 + 0.15 + 0.5 = 1.0
\]
Since the sum of the probabilities equals 1, this condition is also satisfied.

### Conclusion:
Yes, this table represents a valid probability distribution.

Would you like further details or have any questions?

Here are 5 related questions for practice:
1. What is a probability distribution?
2. Can a probability distribution have a negative probability?
3. How do you check if a table represents a probability mass function?
4. What happens if the sum of probabilities is greater than 1?
5. How do continuous probability distributions differ from discrete ones?

**Tip**: Always ensure that the total of all probabilities in any distribution sums to 1!

Determine whether this table represents a probability distribution: X = {7, 3, 1, 8} and P(X) = {0.1, 0.25, 0.15, 0.5}

Learn how to determine whether a table represents a valid probability distribution. The example checks if the sum of the probabilities equals 1 and if all values are non-negative. This problem is suitable for Grades 9-12 students learning about probability.

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