Contextualization

Probability is the chance of something occurring. In our daily lives, we are surrounded by situations where we can apply the logic of probability. Will it rain today? What is the chance of getting a high score on the test? Can we get heads or tails on a coin toss? All these questions are governed by the concept of probability. It is an essential part of mathematics and has vast applications in fields such as statistics, science, engineering, and economics.

To understand the concept of probability well, it is fundamental to learn about probable and improbable events. An event is considered probable if the chance of it occurring is high and improbable if the chance of it happening is low. For example, the probability of rain in the month of January is higher in many places, so we can say that it is a probable event. On the other hand, the chance of a person winning the lottery is very low, making it an improbable event.

Introduction

Mathematics offers a systematic and logical way to understand and describe our world. Probability, as a fundamental part of mathematics, allows us to express the uncertainty of an event in a quantitative way, providing us with tools to make predictions and take informed decisions.

In the field of probability theory, one of the fundamental concepts is the difference between probable and improbable events. That is, understanding and quantifying the chances of an event occurring in a situation of uncertainty. To understand these concepts, it is essential to explore everyday examples and mathematical models, such as rolling a dice or tossing a coin.

This project aims to develop students' understanding of probability and its application in different scenarios. Students are expected to acquire a theoretical and practical understanding of the concepts of probability and apply these concepts in a series of experiments. In addition, they are expected to develop socio-emotional skills, such as time management, communication, problem-solving, creative thinking and proactivity.

Practical Activity

Activity Title

"Discovering the Probable and the Improbable"

Project Objective

The objective of this project is to provide students with a practical and theoretical understanding of the concepts of probability and the difference between probable and improbable events. The students will need to carry out a series of experiments, calculate the probabilities of the events and present their findings in a report.

Detailed Description of the Project

For this project, students will be divided into groups of 3 to 5 people. Each group will carry out two experimental activities to explore probability; one will involve rolling a dice and the other an opinion poll within the school.

In the dice activity, students will roll a dice 100 times and record the results. They will use this data to calculate the probability of each outcome and will discuss whether these events are probable or improbable.

In the survey activity, students will devise a question that can have multiple answers (For example: What is your favorite flavor of ice cream?). They will ask this question to 100 different people in the school and record the results. They will then calculate the probability of each answer and discuss whether each answer is a probable or improbable event.

Materials Needed

• 1 Dice
• Pencil and paper for note taking
• Computer with access to a spreadsheet software (such as Excel)
• Materials for drawing graphs and tables

Detailed Step-by-Step

1. Preparation: Divide the students into groups of 3 to 5 people and provide each group with the materials required for the project.

2. Activity 1 - Dice Experiment:

   • Students roll the dice 100 times and record each roll.
   • They then count and record the number of times each outcome occurred.
   • Using this data, they calculate the probability of each outcome, by dividing the number of times the outcome appeared by the total number of rolls (100).
   • Students discuss whether each outcome is probable or improbable.

3. Activity 2 - Opinion Poll:

   • Students devise a multiple choice question with at least five answer options.
   • They ask this question to 100 different people in the school and record each answer.
   • They then count and record the number of times each answer was given.
   • Using this data, they calculate the probability of each answer, by dividing the number of times the answer was given by the total number of people asked (100).
   • Students discuss whether each answer is a probable or improbable event.

4. Final Report:

   • Students prepare a report that will include an introduction describing the purpose of the project and the relevance of the topic of probability in everyday life.
   • In the development, they should explain in detail the two activities, present the tables and/or graphs with the results obtained and the methodology used to calculate the probabilities.
   • In the conclusion, they should discuss the main learnings and conclusions drawn from the project, highlighting the technical and socio-emotional aspects developed during the work.
   • Finally, they should cite the bibliography used for theoretical and methodological guidance.

This is not only a math project but also involves language skills (for the survey and writing the report), science skills (in designing experiments and interpreting data), and socio-emotional skills (in collaborating, managing time, and communicating). Thus, it is expected to take around 12 hours to complete.

Project Deliverables

Each group must submit:

1. Raw data from the dice and survey experiments.
2. Probability calculations for each event/outcome.
3. Graphs and/or tables representing the results of each activity.
4. A written final report, with introduction, development, conclusions, and bibliography.

This report should be the connection between all parts of the project, offering a detailed view of all the steps from planning to investigation and conclusions. Everything should be detailed and explained in a clear and structured way to allow a complete understanding of the process and the results.