Contextualization

Probability is a mathematical concept that we use to express the chance or possibility of something happening. It is represented by a fraction or percentage and ranges from 0 (when an event is impossible) to 1 (when an event is certain).

There are different types of events in probability, including independent and dependent events. Independent events are those whose occurrence or non-occurrence does not affect the probability of other events happening. On the other hand, dependent events have their probabilities altered depending on the occurrence or non-occurrence of other events.

Understanding the difference between these two types of events is crucial for correctly calculating probabilities, which can be applied to many aspects of our daily lives, from deciding whether to take an umbrella or not based on the weather forecast to more complex issues such as financial investments.

The world around us is full of uncertainties, and probability helps us navigate through these uncertainties. The idea of independent and dependent events is a fundamental aspect in this regard. Consider a medical diagnosis, depending on your symptoms, a doctor can consider the probability of different diseases. Each disease can be considered an event, and each symptom can influence the probability of each disease, making these events dependent on each other. On the other hand, if different diseases have similar symptoms, the presence or absence of a symptom may not alter the probability of another disease, setting up independent events.

Students are encouraged to explore the following reliable resources in Portuguese to deepen their knowledge on the subject:

1. Mathematics Portal - Probability
2. Khan Academy - Introduction to Probability
3. Univesp - Probability Fundamentals

Practical Activity: Independent Events in Card Games

Project Objective

The objective of this activity is to explore and understand independent events through a practical interactive experiment. Students will play a card game to collect data and calculate the probability of independent events.

Detailed Project Description

For this project, students will form groups of 3 to 5 and will play a card game where they will record and calculate the probabilities of different types of events. Students will explore the theory of independent and dependent events, conduct the experiment, record the results, and finally interpret them to draw conclusions.

Required Materials

• A standard deck of 52 playing cards (without jokers)
• Paper and pen for recording results

Detailed Step-by-Step Guide for the Activity

1. Research and Preparation: Start by reviewing probability concepts and independent events. Make sure all group members understand the basic concepts.

2. Experiment Setup: Shuffle the cards and place them face down. The cards will be drawn one by one in each turn.

3. Experiment Execution: In each turn, a group member will draw a card (without returning it to the deck) and record the color of the card (red or black). Repeat this process until all cards are drawn from the deck.

4. Data Collection: Record the sequence of colors drawn. For example, if the first card drawn was red, the second black, the third red, and so on, record: red, black, red, ...

5. Probability Calculation: Based on the recorded sequence, calculate the probability that after drawing a black card, the next card will also be black. Repeat the calculation for the reverse situation (red card followed by another red card).

6. Results Analysis: Discuss the results obtained as a group. Are the events independent or dependent? How was this perceived?

7. Report Writing: Based on the practical activity, each group should produce a report containing:

   a. Introduction: Contextualization of the topic, its relevance, real-world applications, and the objective of this project.

   b. Development: Description of the theory of independent and dependent events, detailed description of the activity performed and methodology used, presentation and discussion of the results obtained.

   c. Conclusion: Recap of the main points of the study, explanation of the learnings obtained, conclusions about independent and dependent events, and about the project's theme in general.

   d. Bibliography: Indicate the sources of information used for the work.

Remember: the report should include the analysis of the collected data, the conclusions drawn, and, very importantly, each group's reflection on what probabilities represent in the real world and how they are used to make decisions based on risks and uncertainties.

Project Delivery

This project will be delivered in two parts: an oral presentation and a written report. For the oral presentation, groups should present the results of their experiment and their conclusions to the class. The written report should follow the format described above, and the references should be in accordance with the school's standard. The report should be submitted one week after the oral presentation.