Math Problem Statement

Fait moi ce devoir

Fait moi ce devoir

Solution

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


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 PJ(F)P_J(F) and PJ(FC)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, PJ(F)P_J(F): PJ(F)=young womentotal young people=4575=0.6P_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 FF, JJ, and CC.
  • Calculate conditional probabilities for gender and participation in competitions.

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=12(GF+EF)IF = \frac{1}{2}(GF + EF)
  • FJ=12(FJ+FB)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 NLMJKINLMJKI 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.


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"

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.

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Math Problem Analysis

Mathematical Concepts

Probability
Conditional Probability
Set Theory (Venn Diagrams)
Vector Geometry
3D Geometry
Platonic Solids

Formulas

P(A|B) = P(A ∩ B) / P(B)
Vector addition (IF = 1/2 * (GF + EF))

Theorems

Symmetry of geometric shapes
Platonic solids theory

Suitable Grade Level

Grades 10-12