Contextualization

Understanding the properties of probability is essential for making accurate predictions and informed decisions in a variety of fields, such as science, engineering, economics, and medicine.

Probability is a quantitative measure that expresses the likelihood of an event occurring. It is expressed by a number between 0 and 1, where 0 indicates that the event will certainly not occur and 1 indicates that the event will certainly occur. Some of the most important properties of probability include additivity, multiplicativity, and normalization.

Additivity tells us that the probability of one event or another event occurring (union of events) is the sum of the probabilities of each event occurring individually if the events are mutually exclusive.

Multiplicativity informs us that the probability of two events happening simultaneously (intersection of events) is the product of the probabilities of each event happening if the events are independent.

Normalization establishes that the sum of all possible probabilities in a sample space is 1. This means that some event must necessarily occur.

These concepts will be explored in depth in our project, so be prepared to delve into them!

Probability is not just a fascinating topic of theoretical study - it has robust and powerful practical applications in our daily lives and in many careers. For example, a scientist can use probability to predict the outcome of an experiment, an engineer can use it to assess the reliability of a system, and an economist can use it to forecast future trends.

Moreover, almost all of us use probability in our daily lives, often without even realizing it. For example, when we check the weather forecast and decide whether to take an umbrella or not, we are considering the probability of rain.

For a deeper exploration in your own time and to assist in completing this project, we suggest the following resources:

• The book "How to Solve It: A New Aspect of Mathematical Method" by George Polya. This book is an excellent introduction to problem-solving strategies, including those that can be approached using probability.

• The Khan Academy website (https://www.khanacademy.org/math/statistics-probability), which covers a wide range of topics in statistics and probability, and provides practical examples.

• The Só Matemática website (https://www.somatematica.com.br/emedio/probabilidade.php) also has a section dedicated to probability, with clear explanations and many examples.

Remember, the goal of this project is to learn to work together to solve complex problems, develop time management skills, and of course, have fun while learning!

Practical Activity

Activity Title: "Probability in Everyday Life"

Project Objective

This activity aims to provide students with the opportunity to understand and apply the properties of probability in a practical way, in a playful and engaging context to encourage collaboration and engagement.

Project Description

Students will be divided into groups of 3 to 5 members, and each group will create a board game that involves the use of probability. The game should be designed in such a way that the probability of certain events occurring affects the progress of the game. In addition, students must include an explanatory manual detailing how the properties of probability are applied in the development of the game.

Required Materials

• Cardboard or card paper for the base of the board
• Colored markers, pens, or pencils to draw on the board
• Small objects to use as markers in the game
• Dice or a roulette to determine the moves
• Computer for research and writing the final report

Step by Step

1. Each group must define the theme and basic rules of the game. This may include the number of players, the sequence of turns, how to win, etc.

2. Once the basic rules are defined, the group must identify where the properties of probability can be applied. For example, the probability of an event can determine the number of spaces a player can move on the board.

3. Using cardboard or card paper, students must create the game board.

4. With markers, pens, or pencils, students will draw on the board the elements that are part of the game.

5. At the end of the game, students will create an explanatory manual, which in addition to explaining how to play, must detail the use of the properties of probability in the game.

6. Each group must test their game to ensure that all rules are working as expected.

7. After completing these steps and making the necessary corrections to any problems found, the groups must present their game to the class and/or to the teacher.

Project Deliverables

The project will be divided into two main parts: the game presentation and the final report submission.

Game Presentation

Each group will have a pre-established time to present their game to the class and explain how the properties of probability were integrated into the project. During the presentation, students must explain the rules of the game, how these rules were developed, and show an example of a game round.

Final Report

After the game presentation, each group must submit a written report that includes:

• Introduction: Students must contextualize the theme, its relevance and real-world application, and the objective of the game. In this case, how probability is used in their game.

• Development: Students need to explain in detail the rules of the game, how the properties of probability were applied, the methodology used for the game development, and present the results obtained, such as the reception of the game by other students and the lessons learned.

• Conclusion: Students should conclude the work by summarizing the main points and explaining the learnings obtained. In this section, they can discuss how the project contributed to their understanding of the properties of probability.

• Bibliography: Students should indicate the sources they used to develop the project, such as books, web pages, videos, etc.

The report will provide each group with the opportunity to reflect on the work done and demonstrate their understanding of the properties of probability in a practical context.