Lesson Note on Probability: Chance Experiments and Mutually Exclusive Events for SSS 2

Class: Senior Secondary School 2 (SS2 / SSS2)
Term: 3rd Term
Week: 1
Age: 16 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Probability
Previous Lesson: .
Topic: PROBABILITY (I)
Subject Matter: Throwing of dice, tossing of coin and pack of playing cards, theoretical and experimental probability, mutually exclusive events.

Specific Objectives

By the end of the lesson, pupils should be able to:

Cognitive Domain:

• Define probability, outcome, event, and sample space.
• Distinguish between theoretical and experimental probability.
• Identify mutually exclusive events.
• Calculate the probability of simple and mutually exclusive events.

Affective Domain:

• Appreciate the importance of probability in everyday life.
• Show interest in solving probability problems.

Psychomotor Domain:

• Perform experiments with coins, dice, and playing cards to determine outcomes.
• Apply the formulas for theoretical and experimental probability correctly.

Social Domain:

• Work cooperatively with peers to explore the properties of chance instruments.

Reference Materials

The following resources were used in planning this lesson:

• Senior Secondary Schools Education Curriculum
• State Unified Scheme of Work
• New General Mathematics for Senior Secondary Schools 2

Instructional Materials

The teacher will teach this lesson with the aid of:

• Ludo dice
• A coin
• A pack of playing cards
• Whiteboard and markers

Rationale for the Lesson

This lesson helps pupils understand the likelihood of events occurring, which is a fundamental concept in mathematics. It enables pupils to apply mathematical reasoning to situations involving chance, which is useful in decision-making and various real-world scenarios.

Prerequisite/Previous Knowledge

Pupils have a basic understanding of fractions, ratios, and simple counting techniques.

Lesson Content/Board Summary

PROBABILITY (I)

Introduction to Probability

Probability is the measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain.

• Experiment: An action or process that leads to one of several possible outcomes. (e.g., tossing a coin, throwing a die).
• Outcome: A single result of an experiment. (e.g., getting a Head when tossing a coin).
• Event: A specific outcome or a set of outcomes of an experiment. (e.g., getting an even number when throwing a die).
• Sample Space (S): The set of all possible outcomes of an experiment. (e.g., for a coin toss, S = {Head, Tail}).

Theoretical Probability

Theoretical probability is the probability of an event based on reasoning and the total possible outcomes, assuming all outcomes are equally likely. It is calculated using the formula:

P(Event A) = (Number of favorable outcomes for A) / (Total number of possible outcomes in the sample space)

Examples:

Example 1: Tossing a Coin

What is the probability of getting a Head when tossing a fair coin?

• Step 1: Identify the sample space. S = {Head, Tail}. Total outcomes = 2.
• Step 2: Identify favorable outcomes for getting a Head. Favorable outcomes = {Head}. Number of favorable outcomes = 1.
• Step 3: Calculate the probability. P(Head) = 1/2.

Example 2: Throwing a Die

What is the probability of getting an even number when throwing a fair six-sided die?

• Step 1: Identify the sample space. S = {1, 2, 3, 4, 5, 6}. Total outcomes = 6.
• Step 2: Identify favorable outcomes for getting an even number. Favorable outcomes = {2, 4, 6}. Number of favorable outcomes = 3.
• Step 3: Calculate the probability. P(Even Number) = 3/6 = 1/2.

Example 3: Drawing a Card

What is the probability of drawing an Ace from a standard pack of 52 playing cards?

• Step 1: Identify the total number of outcomes. Total cards = 52.
• Step 2: Identify the number of favorable outcomes (Aces). There are 4 Aces (Ace of Spades, Hearts, Clubs, Diamonds). Number of favorable outcomes = 4.
• Step 3: Calculate the probability. P(Ace) = 4/52 = 1/13.

Experimental Probability

Experimental probability is based on the results of actual experiments or observations. It is calculated as:

P(Event A) = (Number of times Event A occurs) / (Total number of trials in the experiment)

For instance, if you toss a coin 100 times and get 48 Heads, the experimental probability of getting a Head is 48/100 = 0.48.

Mutually Exclusive Events

Two events are mutually exclusive if they cannot happen at the same time. This means they have no common outcomes.

If A and B are mutually exclusive events, the probability of A or B occurring is:

P(A or B) = P(A) + P(B)

Example: Drawing a Card (Mutually Exclusive)

What is the probability of drawing a King or a Queen from a standard pack of 52 playing cards?

• Step 1: Determine if the events are mutually exclusive. A card cannot be both a King and a Queen at the same time, so they are mutually exclusive.
• Step 2: Calculate the probability of drawing a King. P(King) = 4/52.
• Step 3: Calculate the probability of drawing a Queen. P(Queen) = 4/52.
• Step 4: Add the probabilities. P(King or Queen) = P(King) + P(Queen) = 4/52 + 4/52 = 8/52 = 2/13.

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher greets the pupils, reviews the previous lesson briefly, and then introduces the topic of probability by asking pupils what they understand by “chance” or “likelihood” in everyday situations.
Pupils’ Activity: Pupils respond to the teacher’s questions and share their ideas about chance.
Learning Point: Pupils connect prior knowledge to the new topic of probability.

Step 2: Exploring Instruments of Chance

Time: 8 minutes
Teaching Skill: Demonstration/Observation
Teacher’s Activity: The teacher displays a coin, a die, and a pack of playing cards. The teacher leads pupils to examine these items, asking them to identify the number of faces on the coin and die, and the total number of cards in the pack.
Pupils’ Activity: Pupils examine the coin, die, and cards, identifying the number of faces and the total number of cards.
Learning Point: Pupils become familiar with common instruments used in probability experiments.

Step 3: Understanding Outcomes and Sample Space

Time: 7 minutes
Teaching Skill: Activity/Explanation
Teacher’s Activity: The teacher asks pupils to take turns tossing the coin or throwing the die and noting the outcomes. The teacher then defines key terms like experiment, outcome, event, and sample space, using the coin and die examples.
Pupils’ Activity: Pupils participate in the coin toss and die throw, record outcomes, and listen to the definitions.
Learning Point: Pupils understand the basic terminology of probability.

Step 4: Theoretical Probability

Time: 8 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher explains theoretical probability, its formula, and guides pupils through worked examples involving the coin, die, and playing cards, writing the solutions on the board.
Pupils’ Activity: Pupils listen, ask questions, and copy the definitions and examples into their notebooks.
Learning Point: Pupils learn how to calculate theoretical probability for simple events.

Step 5: Experimental Probability

Time: 5 minutes
Teaching Skill: Explanation/Comparison
Teacher’s Activity: The teacher explains experimental probability and its formula, contrasting it with theoretical probability by discussing how results might differ in real trials versus ideal calculations.
Pupils’ Activity: Pupils listen and understand the difference between theoretical and experimental probability.
Learning Point: Pupils understand that probability can be determined through actual experiments.

Step 6: Mutually Exclusive Events

Time: 7 minutes
Teaching Skill: Explanation/Illustration
Teacher’s Activity: The teacher defines mutually exclusive events and provides the formula for calculating their probability. The teacher then works through an example using the pack of playing cards (e.g., drawing a King or a Queen).
Pupils’ Activity: Pupils listen, ask clarifying questions, and copy the definition and example into their notes.
Learning Point: Pupils identify and calculate probabilities for mutually exclusive events.

Step 7: Evaluation/Review

Time: 5 minutes
Teaching Skill: Questioning/Assessment
Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. Define probability in your own words.
2. List the sample space when throwing a single die.
3. What is the theoretical probability of getting a ‘2’ when throwing a fair die?
4. Distinguish between theoretical and experimental probability.
5. Are “drawing a red card” and “drawing a black card” from a pack mutually exclusive events? Explain why.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 2 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key learning points on probability, theoretical vs. experimental probability, and mutually exclusive events. The teacher then assigns relevant homework from the textbook.
Pupils’ Activity: Pupils listen to the summary and copy down the homework assignment.
Learning Point: Pupils consolidate their understanding and prepare for independent practice.

Lesson Keywords

• Probability – The measure of the likelihood that an event will occur.
• Outcome – A single result of an experiment.
• Event – A specific outcome or a set of outcomes of an experiment.
• Sample Space – The set of all possible outcomes of an experiment.
• Theoretical Probability – Probability based on reasoning and total possible outcomes.
• Experimental Probability – Probability based on results of actual experiments.
• Mutually Exclusive Events – Events that cannot happen at the same time.
• Coin – A flat, typically round piece of metal used for currency, often used in probability experiments.
• Die – A small cube with faces marked with spots, used in games of chance.
• Playing Cards – A pack of cards used for games, typically 52 in number.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide more complex probability problems involving multiple dice or combinations of events. For pupils needing more support, the teacher will provide additional guided practice with simpler examples and visual aids, ensuring they understand the basic definitions and formulas before moving to more complex problems.

Note for teachers using this lesson plan

Ensure that all instructional materials (coin, die, pack of cards) are readily available and in good condition for practical demonstration. Encourage pupil participation in the activities to make the learning more engaging and concrete. Emphasize the connection between the classroom examples and real-life situations where probability is applied.