Contextualization

Probability is a widely used mathematical tool in many areas, including Physics, Engineering, Social Sciences, Computer Science, Economics, and Medicine. It allows us to quantify the uncertainty associated with events, making it possible to make predictions and informed decisions even when we do not have all the information.

In this project, we will focus on complementary events, which are fundamental concepts in probability theory. A complementary event is the set of all outcomes that are not part of the original event. The sum of the probabilities of an event and its complementary event is always equal to 1, as one of these must certainly occur.

Introduction

The probability of an event is the degree of certainty that this event will occur, based on the available information. It is generally expressed as a number between 0 and 1. The probability of an impossible event is 0, while the probability of a certain event is 1.

Complementary events are especially useful when we want to calculate the probability of an event not occurring. In many cases, it is easier to calculate the probability of the complementary event and then subtract that result from 1.

Let's illustrate this with a simple example: Suppose we have a fair coin, and we want to calculate the probability of not getting heads. We know that the probability of getting heads is 0.5 (as we have two equally likely possibilities: heads or tails). So, the probability of the complementary event (not getting heads, i.e., getting tails) is 1 - 0.5 = 0.5.

Practical Activity

Activity Title: The Probability Game - Understanding Complementary Events

Project Objective

The main objective of this project is for you, in groups of 3 to 5 participants, to explore the concept of complementary events in probability through a card game. By the end of the project, you should be able to understand and calculate the probabilities of complementary events.

Detailed Project Description

In this activity, each group will create a card game with clear rules and objectives, incorporating the concept of complementary events in probability. They must also create a detailed report explaining the development and results of the project.

Required Materials

• Deck of cards
• Paper and pen for notes

Detailed Step-by-Step for Activity Execution

Phase 1: Game Planning

1. Start the activity by discussing the game rules as a group. The game should be designed to include situations where you need to calculate the probability of complementary events.
2. Define the rules clearly and concisely.

Phase 2: Game in Action

3. Play the game several times and record the results after each round.
4. During the gameplay, calculate the probability of different events and their complementary events based on the recorded results.

Phase 3: Results Analysis

5. After playing enough rounds, analyze the results. Calculate the probability of each event and its complementary event in the game.
6. Compare the results obtained with the theoretically expected results.

Project Deliverables

Groups should deliver, at the end of one month, a written document containing the following topics:

1. Introduction

Provide context to the project by describing the concept of probability and complementary events. Explain the relevance of this concept in various real-life applications.

2. Development

Detail the theory of complementary events in probability. Describe the game created by the group, explaining the rules and the reason they were established in this way. Explain the methodology used to record the game results and calculate the probabilities. Present the results obtained, discussing and comparing them with the theoretically expected results.

3. Conclusion

Conclude the work by summarizing the main points of the project. Explain the learnings obtained from the project and how the experience of creating and playing the game helped in better understanding the concept of complementary events in probability.

4. Bibliography

List all resources used for the project, including books, websites, videos, among others. Make sure to follow the appropriate citation format for all resources.