Objectives (5 - 7 minutes)

1. Understand the concept of dependent events and its application in probability.
   • Define what dependent events are.
   • Explain how the occurrence of one event can affect the probability of occurrence of another event.
2. Apply the formula for the probability of dependent events.
   • Develop the skill to calculate the probability of occurrence of dependent events.
3. Solve practical problems involving dependent events.
   • Develop the skill to apply theoretical knowledge in practical situations.
   • Reinforce the understanding of the concept through concrete examples.

Secondary objectives:

• Encourage active participation of students through discussions and group problem-solving.
• Develop critical thinking and problem-solving skills.
• Stimulate the application of mathematics in everyday life, demonstrating the relevance and usefulness of the discipline.

Introduction (10 - 15 minutes)

1. Review of previous concepts:

   • The teacher begins the class by briefly recalling the concepts of probability and independent events, which were covered in previous classes.
   • It is important that students have a solid understanding of these concepts to facilitate the introduction of the new topic.

2. Presentation of problem situations:

   • The teacher presents two problem situations that will serve as scenarios for the introduction of the topic of dependent events:
     • Situation 1: There are 5 red balls and 3 blue balls in a bag. If a ball is drawn at random and not replaced, what is the probability of drawing a blue ball on the second attempt?
     • Situation 2: A deck of 52 cards is shuffled, and two cards are drawn at random. What is the probability that the second card is a king given that the first card drawn was a king?

3. Contextualization of the subject:

   • The teacher emphasizes the importance of studying dependent events, showing how this concept is applied in various areas, such as statistics and gambling.
   • The teacher can also bring examples from everyday life, such as calculating the probability of certain events occurring in sequence, to show students how mathematics is present in everyday life.

4. Introduction of the topic:

   • The teacher introduces the topic of dependent events, explaining that unlike independent events, the occurrence of one event can influence the probability of occurrence of another event.
   • To spark students' interest, the teacher can share curiosities or stories related to the topic. For example, they could talk about how the probability of winning the lottery is affected by the order of the numbers drawn, or how the probability of hitting a target with an arrow is affected by the position of the previous arrows.

Development (20 - 25 minutes)

1. Explanation of the theory (10 - 12 minutes)

   • The teacher presents the definition of dependent events, explaining that they are events whose probability of occurrence is affected by the occurrence of another event.
   • Next, the teacher explains how to calculate the probability of dependent events. They can use the following formula: P(A and B) = P(A) x P(B|A), where P(A and B) is the probability of events A and B occurring together, P(A) is the probability of event A occurring, and P(B|A) is the probability of event B occurring given that event A occurred.
   • The teacher should explain each element of the formula and how to calculate the probability of occurrence of each event.
   • It is important for the teacher to distinguish between P(A and B) and P(A or B) so that students do not confuse the two concepts.

2. Solving the examples (5 - 7 minutes)

   • The teacher solves the problem situations presented in the Introduction, using the theory explained. They should show step by step how to calculate the probability of occurrence of each event and the probability of occurrence of the two events together.
   • During the solution, the teacher should explain each step clearly and in detail so that students can understand the calculation process.
   • It is important for the teacher to encourage students to ask questions and participate in the discussion to ensure that everyone is following and understanding the content.

3. Group discussion (5 - 6 minutes)

   • The teacher divides the class into groups and gives each group a problem situation to solve. The problem situations should involve the probability of occurrence of dependent events.
   • The teacher circulates around the room, guiding the groups and clarifying doubts. It is important for the teacher to encourage students to discuss their solution strategies and explain their reasoning so that they can learn from each other.
   • After a set time, each group presents its solution to the class. The teacher corrects and supplements the answers, reinforcing the concepts learned.

4. Individual practice (2 - 3 minutes)

   • The teacher gives a set of exercises for students to solve individually. The exercises should involve the probability of occurrence of dependent events.
   • The teacher circulates around the room, observing and assisting students. They should correct the exercises on the spot so that students can see their mistakes and correct their difficulties immediately.
   • It is important for the teacher to encourage students to do the exercises even if they have doubts, as practice is essential for learning. In addition, the teacher should praise and encourage students when they get the exercises right to motivate them to continue studying.

Feedback (8 - 10 minutes)

1. Group discussion (3 - 4 minutes)

   • The teacher brings the class together and asks each group to share the solutions or conclusions they reached when solving the problem situations.
   • The teacher should ensure that all students have the opportunity to speak and all groups have a chance to present.
   • During the presentations, the teacher should ask questions to stimulate critical thinking and to check if the students understood the concepts discussed.

2. Connection with the theory (2 - 3 minutes)

   • After the presentations, the teacher should do a brief review of the theoretical concepts discussed in the lesson.
   • The teacher can highlight the connections between the theory and the problem situations solved, to show students how to apply the concepts learned in different contexts.
   • The teacher can also revisit the formula for the probability of dependent events, explaining each element again and how to calculate the probability of occurrence of dependent events.

3. Individual reflection (2 - 3 minutes)

   • The teacher asks students to reflect individually on what they learned in the lesson.
   • The teacher asks guiding questions, such as: "What was the most important concept you learned today?" and "What questions have not yet been answered?"
   • The teacher gives the students a minute to think about the questions and then asks a few students to share their answers with the class.
   • The teacher should listen carefully to the students' answers and ask any necessary questions to clarify any confusion or misunderstandings.

4. Closing (1 minute)

   • The teacher ends the lesson by reinforcing the importance of studying dependent events and probability in everyday life.
   • The teacher also informs the students about the content of the next lesson and any homework or reading that needs to be done before the next lesson.
   • Finally, the teacher thanks the students for their participation and encourages them to continue studying and practicing the concepts learned.

Conclusion (5 - 7 minutes)

1. Recapitulation of the main points (1 - 2 minutes)

   • The teacher begins the Conclusion by recalling the fundamental concepts that were covered in the lesson. They give a brief review of what dependent events are, how to calculate the probability of dependent events, and the difference between dependent and independent events.
   • To reinforce learning, the teacher can conduct a quick oral quiz, asking students to recall and explain the concepts learned.

2. Connection between theory, practice, and applications (1 - 2 minutes)

   • The teacher highlights how the lesson connected theory, practice, and applications. They emphasize how the theory of dependent events was applied in solving problem situations and interpreting practical examples.
   • The teacher also reinforces how understanding this concept is fundamental in various areas, such as statistics, gambling, and predicting everyday events.

3. Supplementary materials (1 minute)

   • The teacher suggests some supplementary study materials for students who want to deepen their knowledge of dependent events. These materials may include math textbooks, educational websites, explanatory videos, and interactive probability games.
   • For example, the teacher could recommend reading specific chapters of a math textbook, exploring websites like Khan Academy, which offers video lessons on probability, and watching videos on YouTube that demonstrate the application of probability in everyday situations in a playful way.

4. Importance of the topic (1 - 2 minutes)

   • Finally, the teacher ends the lesson by emphasizing the importance of studying dependent events and probability. They explain how these concepts are useful for understanding and predicting various phenomena in everyday life, from the probability of winning the lottery to the probability of an accident on a busy road.
   • The teacher also highlights how the ability to calculate and interpret the probability of dependent events can be valuable in various careers, such as statistics, finance, engineering, and medicine.

5. Closing (1 minute)

   • The teacher thanks the students for their participation and effort during the lesson. They encourage the students to continue studying and practicing the concepts learned, and they emphasize that they are available to clarify any doubts.
   • Finally, the teacher wishes everyone a good week and says goodbye, ending the class.