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The image contains a probability problem involving balls drawn from a box with 5 blue, 2 green, and 3 red balls. Here’s a detailed breakdown of the problem and its sub-questions:

**Problem Statement:**
Ten balls (with different colors) are selected at random from a box that contains 5 blue, 2 green, and 3 red balls. Let \( X \) be the number of blue balls selected, and \( Y \) be the number of green balls selected. 

Given that:
- The total number of balls is \( 5 + 2 + 3 = 10 \).

**Sub-questions:**
1. **(a)** Find the marginal distribution of \( X \) and \( Y \).
2. **(b)** Find the conditional distribution of \( X \) given that \( Y = 1 \), and use it to determine \( P(X = 2 \mid Y = 1) \).
3. **(c)** Show that the random variables \( X \) and \( Y \) are not statistically independent.

### Solution Outline:

**1. (a) Marginal Distribution of \( X \) and \( Y \):**

Let’s define the marginal distribution of \( X \) and \( Y \) individually.

- **\( X \):** Represents the number of blue balls drawn, which can be any value from 0 to 5.
- **\( Y \):** Represents the number of green balls drawn, which can be any value from 0 to 2.

These distributions can be calculated using combinations of selecting specific numbers of each colored ball from the total.

**2. (b) Conditional Distribution \( P(X \mid Y = 1) \):**

This involves finding the probability distribution of \( X \) given that exactly 1 green ball is selected (\( Y = 1 \)). The specific probability \( P(X = 2 \mid Y = 1) \) can be calculated using the formula for conditional probability.

**3. (c) Independence of \( X \) and \( Y \):**

To show \( X \) and \( Y \) are not independent, verify if \( P(X, Y) = P(X)P(Y) \). If this equation does not hold for all pairs \((X, Y)\), the variables are dependent.

Would you like detailed step-by-step calculations for each part of the problem? Let me know if you have any specific questions or need further clarifications!

### Related Questions:
1. How do you compute the marginal distribution of a discrete random variable?
2. What are the steps to calculate conditional probability?
3. How can you determine if two random variables are statistically independent?
4. What role does the combination formula play in probability distributions?
5. How do changes in sample space affect the marginal and conditional probabilities?

**Tip:** Always clearly define your random variables and conditions before calculating probabilities to avoid confusion during problem-solving.

Ten balls are selected at random from a box that contains 5 blue, 2 green, and 3 red balls. Let X be the number of blue balls selected, and Y be the number of green balls selected. The problem involves finding the marginal distribution of X and Y, the conditional distribution of X given that Y = 1, and proving that the random variables X and Y are not statistically independent.

Explore the problem of selecting balls at random from a box containing blue, green, and red balls. This solution covers key concepts such as marginal distribution, conditional probability, and statistical independence, ideal for undergraduate students studying advanced probability and statistics.

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