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.