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### Translation of the Problem:

**Exercise 2:**

1. Let \( n \) be a nonzero natural number. In a probability space \((\Omega, A, P)\), \( A \) and \( B \) are two events with nonzero probabilities.  
   - Assume \( P_B(A) \leq P(A) \). Compare \( P_A(B) \) and \( P(B) \).  
   
   (Here \( P_B(A) \) is the conditional probability of \( A \) given \( B \), and \( P_A(B) \) is the conditional probability of \( B \) given \( A \)).

2. A box contains indistinguishable balls numbered from \( 1 \) to \( 2n \). Benoît draws a ball randomly, without replacement, and then Gilles draws another ball.  
   - Gilles gets a number between \( n+1 \) and \( 2n \). Is the probability that Benoît drew a number between \( n+1 \) and \( 2n \) **greater than**, **equal to**, or **less than** \( \frac{1}{2} \)?

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### Solution Approach:

#### 1. **Conditional Probabilities:**

   - Given \( P_B(A) \leq P(A) \):
     \[
     P_B(A) = \frac{P(A \cap B)}{P(B)}, \quad \text{and } P(A)
     \]
     Compare \( P_A(B) = \frac{P(A \cap B)}{P(A)} \) with \( P(B) \).  
     Simplify expressions to establish a relationship between \( P_A(B) \) and \( P(B) \).

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#### 2. **Balls Problem:**

   - Total balls: \( 2n \).
   - Probability for Gilles:
     \[
     \text{Gilles draws a number in } (n+1, 2n] \implies \text{these are } n \text{ numbers}.
     \]
     Analyze probabilities under the following conditions:
     - Benoît draws first and Gilles' range affects the outcomes.
     - Compare probabilities for Benoît's numbers.

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Exercise 2: Let n be a nonzero natural number. In a probability space (Ω, A, P), A and B are two events with nonzero probabilities. 1. Assume P_B(A) ≤ P(A). Compare P_A(B) and P(B). 2. A box contains indistinguishable balls numbered from 1 to 2n. Benoît draws a ball randomly without replacement, then Gilles draws another ball. Gilles gets a number between n+1 and 2n. Is the probability that Benoît drew a number between n+1 and 2n greater than, equal to, or less than 1/2?

Explore solutions for a probability problem involving conditional probabilities and combinatorics. Learn how to compare P_A(B) and P(B), and analyze the probabilities of drawing balls from a numbered set.

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