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Let's solve the probabilities based on the given probability distribution table. Here is the table and the probabilities to find:

| \( x \) | \( P(x) \) |
|---------|------------|
| 0       | 0.24       |
| 1       | 0.28       |
| 2       | 0.03       |
| 3       | 0.37       |
| 4       | 0.08       |

### Calculations

1. **\( P(x < 2) \)**:
   - This represents \( P(x = 0) + P(x = 1) \).
   - \( P(x < 2) = 0.24 + 0.28 = 0.52 \).

2. **\( P(x \leq 2) \)**:
   - This represents \( P(x = 0) + P(x = 1) + P(x = 2) \).
   - \( P(x \leq 2) = 0.24 + 0.28 + 0.03 = 0.55 \).

3. **\( P(x > 1) \)**:
   - This represents \( P(x = 2) + P(x = 3) + P(x = 4) \).
   - \( P(x > 1) = 0.03 + 0.37 + 0.08 = 0.48 \).

4. **\( P(x \geq 1) \)**:
   - This represents \( P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4) \).
   - \( P(x \geq 1) = 0.28 + 0.03 + 0.37 + 0.08 = 0.76 \).

### Summary of Answers:
- \( P(x < 2) = 0.52 \)
- \( P(x \leq 2) = 0.55 \)
- \( P(x > 1) = 0.48 \)
- \( P(x \geq 1) = 0.76 \)

Would you like more details on any of these calculations? 

Here are five related questions you might find helpful:
1. What is the probability that \( x = 3 \)?
2. How can we verify that this is a valid probability distribution?
3. What is the probability of \( x \) being either 0 or 4?
4. What is the expected value of \( x \) in this distribution?
5. How would the probabilities change if \( P(x = 4) \) increased?

**Tip:** To ensure it's a valid probability distribution, the sum of all probabilities should equal 1.

Use the probability distribution table to find the specified probabilities:
- P(x < 2)
- P(x ≤ 2)
- P(x > 1)
- P(x ≥ 1)

Learn how to calculate the probabilities for events like P(x < 2) and P(x ≥ 1) from a discrete probability distribution table. This problem includes concepts such as summing probabilities for ranges of x-values and using basic probability laws. Suitable for students in Grades 10-12.

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