Objectives (5 - 7 minutes)

1. To introduce the concept of conditional probability to the students and explain its importance in mathematical scenarios and real-life situations.
2. To demonstrate how conditional probability works through various examples, ranging from simple to complex situations.
3. To encourage students to engage in problem-solving activities related to conditional probability, enhancing their comprehension and application skills.

Secondary Objectives:

• To foster an interactive learning environment where students feel comfortable asking questions and discussing the topic.
• To promote a collaborative learning approach by encouraging students to help each other understand and solve conditional probability problems.
• To instill an appreciation for the relevance and applicability of conditional probability in various fields such as science, business, and everyday decision-making.

Introduction (10 - 15 minutes)

1. The teacher starts the lesson by reminding students of the basic concept of probability they have learned previously. The teacher briefly revisits the concept of probability as the measure of the likelihood that an event will occur, represented as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. (3 minutes)
2. The teacher then presents two problem situations to serve as starters for the development of the theory that follows:
   • A deck of cards scenario: If a card is drawn from a deck and it is found to be a heart, what is the probability that it is a queen?
   • A dice rolling scenario: If a die is rolled and the result is an even number, what is the probability that it is a 4? (4 minutes)
3. The teacher then contextualizes the importance of conditional probability with real-world applications. They explain that conditional probability is used in a variety of fields such as weather forecasting, insurance, medical diagnoses, and even in our daily decision-making processes. For instance, if it is known that it is raining, the probability that someone will carry an umbrella increases. This is an everyday example of conditional probability. (3 minutes)
4. To grab the students' attention, the teacher introduces the topic with two interesting facts:
   • The Monty Hall problem: This is a probability puzzle based on a game show where contestants had to choose between three doors, behind one of which was a car and behind the others, goats. The host, knowing what was behind each door, would open one of the doors revealing a goat. The contestant then had to decide whether to stick with their original choice or switch to the remaining unopened door. Surprisingly, the probability of winning the car is higher if the contestant decides to switch. This counter-intuitive result is a classic example of conditional probability.
   • The use of conditional probability in machine learning: Conditional probability is a fundamental concept in machine learning algorithms, especially in natural language processing. For instance, in predictive text input, the next word suggestion is based on the conditional probability of the word given the preceding words. (5 minutes)

Development (20 - 25 minutes)

Activity 1: 'Fruit Salad Forecast'

1. The teacher divides the class into groups of four and provides each group with a bowl containing a mix of different fruits (apples, oranges, and bananas).

2. Each group selects a 'weather forecaster' (one student in each group). The 'forecaster' stays blindfolded while the other students (let's call them 'producers') drop one fruit at a time into a separate bowl. The 'forecaster' should predict the next fruit in line based on the frequency of fruits dropped into the new bowl.

3. Producers write down the sequence of fruits they drop into the new bowl.

4. The 'forecaster' has to predict what fruit will be dropped into the bowl next after hearing what fruit was previously dropped. 'Forecasters' make predictions based on the understanding that the probability of the next fruit depends on the fruit dropped previously (this is the basic principle of conditional probability).

5. At the end of the activity, discuss the predictions and the corresponding theories with the whole class. This helps students to tie the activity back to the concept of conditional probability.

Activity 2: 'Card Sharks'

1. The teacher provides each group with a deck of cards and ask them to shuffle the deck constantly.

2. Each group then draws five cards from the deck. The mission of each group is to find out the conditional probability of drawing a red card given that a card drawn is a face card (Face cards are Jack, Queen, and King).

3. Each group discusses and calculates the conditional probability discussed in step 2. After the groups find the conditional probability, they reshuffle and redraw cards, repeating the process.

4. After several iterations, the groups compare their results and discuss if the calculated probabilities match the theory they've been taught.

Activity 3: 'The Predictive Text Challenge'

1. The teacher prepares a list of common phrases and divides the class into pairs.

2. Each pair writes a phrase from the list in a popular text app with predictive text (like Google Docs or any smartphone).

3. The students then observe the three-word suggestions based on the initial phrase and predict the rest of the phrase using only these suggestions.

4. The goal is to predict the phrase with the minimum number of cues. This exercise helps to explain how conditional probability is at work in our day-to-day technology usage. The better they predict, the faster they understand the concept of conditional probability.

Feedback (10 - 15 minutes)

1. The teacher starts the feedback session by asking each group to present the conclusions they reached during the development activities. The groups explain how they applied the concept of conditional probability in their tasks and share their findings with the rest of the class. The teacher guides the discussion to ensure the students' conclusions align with the theory. (3 minutes)

2. Following the presentations, the teacher opens the floor for a class-wide discussion. Students are encouraged to ask questions about the presentations or provide constructive feedback. The teacher uses this opportunity to correct any misconceptions and clarify any aspects of conditional probability that are still unclear. (4 minutes)

3. The teacher then transitions to an individual reflection period. The students are asked to silently reflect on what they have learned from the lesson and the group activities. They are asked to think about the most important concept they learned and any questions they still have about conditional probability. (1 minute)

4. After the reflection period, the teacher initiates a class discussion based on the reflection prompts. Students share the key concepts they identified and ask any lingering questions. This helps the teacher gauge the level of understanding among the students and identify any areas that may need further explanation in future lessons. (2 minutes)

5. The teacher then provides a summary of the lesson, highlighting the importance of conditional probability in mathematical scenarios and real-life situations. The teacher also emphasizes the applicability of the concept in various fields, reinforcing the relevance of what they have learned. (3 minutes)

6. Finally, the teacher assigns homework based on the lesson. The homework consists of additional problems on conditional probability for students to solve independently. This allows students to practice applying the concept of conditional probability on their own, thereby reinforcing their understanding of the topic. The teacher assures students that they will go over the homework in the next class to clarify any difficulties. (2 minutes)

7. The teacher concludes the lesson by thanking the students for their active participation and encouraging them to continue engaging in the learning process. (1 minute)

Conclusion (5 - 7 minutes)

1. The teacher begins the conclusion by summarizing the main contents of the lesson: The concept of conditional probability, its calculation, and its importance in predicting outcomes based on prior knowledge or events. The teacher reminds the students that conditional probability is the probability of an event given that another event has occurred. The teacher also revisits the activities conducted during the lesson and the key takeaways from each activity. (1 - 2 minutes)

2. The teacher then explains how the lesson connected theory, practice, and applications. The teacher emphasizes that the activities such as 'Fruit Salad Forecast', 'Card Sharks' and 'The Predictive Text Challenge' were designed to help students apply the theoretical knowledge of conditional probability to practical scenarios. The teacher further points out that these activities demonstrated real-world applications of conditional probability, such as in weather forecasting, card games, and predictive text technology. (1 - 2 minutes)

3. Next, the teacher suggests additional materials for students to further their understanding of conditional probability. This could include recommended textbooks, online resources, and interactive math games that focus on conditional probability. The teacher also encourages students to explore real-life scenarios where conditional probability is relevant, such as in sports predictions, medical diagnostics, and stock market trends. (1 minute)

4. Finally, the teacher concludes by explaining the importance of the lesson's topic for everyday life. The teacher emphasizes that conditional probability is not just a mathematical concept, but a tool that helps us make informed decisions based on the knowledge of prior events. From deciding to carry an umbrella based on weather forecasts, to predicting text in our smartphones, conditional probability is an integral part of our daily lives. The teacher encourages the students to apply what they learned in their own lives and to keep exploring the fascinating world of probabilities. (1 - 2 minutes)