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

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.