https://file.aiquickdraw.com/mathbot/file/17284773300125pidxty4.jpg

Let's break down the exercises in the image one by one.

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### **Exercice 2**:
This problem involves a sports club with 250 members, and it asks you to calculate probabilities related to gender (F: female), age (J: young person), and competition participation (C: competitor). Here's how we can address each part:

#### Part 1:
- **Determine the probabilities \( P_J(F) \) and \( P_J(F \cap C) \)**.

We are given:
- 115 women in total,
- 75 young people,
- 45 young women.

- Probability that a randomly selected young person is a woman, \( P_J(F) \):
  \[
  P_J(F) = \frac{\text{young women}}{\text{total young people}} = \frac{45}{75} = 0.6
  \]

#### Part 2:
- **Among the women, there are 50 competitors**:
  - Total women = 115.
  - Competitors = 50 (45 young competitors and 25 older women competitors).

Next, you are asked to use conditional probabilities and a Venn diagram to represent the data.

**Solution for other questions**: You will need to:
  - Construct a Venn diagram with the sets \( F \), \( J \), and \( C \).
  - Calculate conditional probabilities for gender and participation in competitions.

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### **Exercice 3**:
This exercise involves vectors in 3D geometry and a cube structure.

#### Part A:
You are asked to show vector relationships using the symmetry of the cube:
- \( IF = \frac{1}{2}(GF + EF) \)
- \( FJ = \frac{1}{2}(FJ + FB) \)
  
This follows from the fact that the cube's centers create symmetrical relationships between vectors.

#### Part B:
- Justifying that all edges of the octahedron \( NLMJKI \) have the same length: This is based on the symmetry of the cube.
  
#### Part C:
Research on "Platonic solids": The octahedron is a Platonic solid, and Plato was a philosopher who theorized about the geometric shapes.

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### **Exercice 4**:
In this task, you are asked to propose a topic for your "Grand Oral" exam.

You could choose topics like:
- "The application of geometry in real-life scenarios"
- "How probability theory can be applied to decision-making"
- "The role of mathematical models in sports statistics"

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Do you need further details on any of the exercises? 

Here are 5 related questions to expand on the information:
1. What is the formula for conditional probability, and how would you apply it here?
2. How can you draw a Venn diagram to best represent the overlapping sets in Exercice 2?
3. Can you explain the geometric properties of the octahedron in Exercice 3 in more detail?
4. How does vector addition help in solving Part A of Exercice 3?
5. What are other examples of Platonic solids and their properties?

**Tip**: When solving geometry problems involving symmetry, always look for ways to simplify calculations by identifying equal lengths or angles.

Explore probability calculations involving set theory, conditional probabilities, and Venn diagrams in Exercice 2, and vector-based geometry problems in Exercice 3. Learn about Platonic solids and the symmetry of the cube in Exercice 4, ideal for students in Grades 10-12.

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