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### Task 1: Stepwise Calculations for Probability Questions

Given the table of probabilities:

| \( x \) | \( P(x) \) |
|--------|-------------|
| 1      | 0.15        |
| 2      | 0.35        |
| 3      | 0.40        |
| 4      | 0.10        |

We need to answer the following questions:

#### 1. What is the probability that \( P(X > 2) \)?

To calculate the probability that \( X \) is greater than 2, we sum the probabilities of \( X = 3 \) and \( X = 4 \):

\[ P(X > 2) = P(X = 3) + P(X = 4) \]

Step-by-step calculation:
\[ P(X = 3) = 0.40 \]
\[ P(X = 4) = 0.10 \]
\[ P(X > 2) = 0.40 + 0.10 = 0.50 \]

Therefore, the probability that \( X \) is greater than 2 is 0.50.

#### 2. What is the probability that \( P(X = 4) \)?

To find \( P(X = 4) \), we use the probability given in the table for \( X = 4 \):

\[ P(X = 4) = 0.10 \]

Therefore, the probability that \( X \) equals 4 is 0.10.

#### 3. What is the probability that \( P(1 < X < 3) \)?

To find the probability that \( 1 < X < 3 \), we sum the probabilities of \( X = 2 \):

\[ P(1 < X < 3) = P(X = 2) \]

Step-by-step calculation:
\[ P(X = 2) = 0.35 \]

Therefore, the probability that \( X \) is greater than 1 and less than 3 is 0.35.

### Summary of Results
1. \( P(X > 2) = 0.50 \)
2. \( P(X = 4) = 0.10 \)
3. \( P(1 < X < 3) = 0.35 \)

Would you like more details or have any questions?

Here are five more questions you can ask:
1. How do you calculate the expected value (mean) of a probability distribution?
2. What is the variance and how is it calculated for a probability distribution?
3. Can you explain the concept of a cumulative distribution function (CDF)?
4. How do you calculate the probability of multiple independent events occurring together?
5. What are the differences between discrete and continuous probability distributions?

**Tip:** Always make sure the probabilities in a discrete probability distribution table sum up to 1 to ensure it is a valid distribution.

Explore stepwise calculations for probability questions using a given probability distribution table. Understand how to calculate probabilities such as P(X > 2), P(X = 4), and P(1 < X < 3). Suitable for university-level students studying probability theory.

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