Class: Senior Secondary School 2 (SS2 / SSS 2)
Term: First Term
Week: 6
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Probability
Previous Lesson: Polynomials: Roots of Cubic Equation and Relationships.
Topic: PROBABILITY
Subject Matter: Meaning of probability (measure of chance of an event), classical approach (equally likely outcomes), frequential approach (relative frequency from experiments), axiomatic approach (probability rules), sample space, event space, mutually exclusive events, independent events, conditional events

Specific Objectives

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

Cognitive Domain:

Define probability and identify its three main approaches.
List the sample space and event space for simple experiments.
State the rules for mutually exclusive, independent, and conditional events.

Affective Domain:

Appreciate the importance of probability in making predictions and decisions.
Show interest in solving probability problems.

Psychomotor Domain:

Calculate probabilities using the classical and frequential approaches.
Apply the axiomatic rules to solve probability problems.
Differentiate between mutually exclusive, independent, and conditional events.

Collaborate with peers to solve probability-related tasks.
Communicate their understanding of probability concepts clearly.

Reference Materials

The following resources were used in planning this lesson:

Senior Secondary Schools Education Curriculum
State Unified Scheme of Work
New Further Mathematics for Senior Secondary Schools by P.N. Okeke
https://www.example.com/further-maths-probability
https://www.example.com/probability-concepts

Instructional Materials

The teacher will teach this lesson with the aid of:

Ludo dice
Coin
Pack of cards

Rationale for the Lesson

This lesson helps pupils understand the likelihood of events occurring, which is important for making informed decisions in various aspects of life, including games, science, and economics. It provides a foundation for more advanced statistical analysis.

Prerequisite/Previous Knowledge

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

Lesson Content/Board Summary

PROBABILITY

Meaning of Probability

Probability is a numerical measure of the likelihood or chance that an event will occur. Its value ranges from 0 (impossible event) to 1 (certain event).

Approaches to Probability

There are three main approaches to defining and calculating probability:

Classical Approach: This applies when all possible outcomes of an experiment are equally likely.
Formula: P(A) = (Number of favorable outcomes for A) / (Total number of possible outcomes)

Example: The probability of getting a ‘3’ when rolling a fair six-sided die is 1/6, because there is one favorable outcome (3) out of six possible outcomes (1, 2, 3, 4, 5, 6).
Frequential Approach (Empirical/Statistical Probability): This is based on observations from experiments or historical data.
Formula: P(A) = (Number of times event A occurred) / (Total number of trials)

Example: If a coin is tossed 100 times and heads appears 55 times, the frequential probability of getting a head is 55/100 = 0.55.
Axiomatic Approach: This defines probability based on a set of axioms (rules) that probabilities must satisfy.
Axiom 1: The probability of any event A is between 0 and 1, i.e., 0 ≤ P(A) ≤ 1.

Axiom 2: The probability of the sample space S (certain event) is 1, i.e., P(S) = 1.

Axiom 3: For mutually exclusive events A and B, P(A or B) = P(A) + P(B).

Key Terms in Probability

Sample Space (S): The set of all possible outcomes of a random experiment.
Example: For rolling a die, S = {1, 2, 3, 4, 5, 6}.

Example: For tossing two coins, S = {HH, HT, TH, TT}.
Event Space (E): A subset of the sample space, representing a specific outcome or set of outcomes.
Example: For rolling a die, if event A is “getting an even number”, E = {2, 4, 6}.

Types of Events

Mutually Exclusive Events: Two events A and B are mutually exclusive if they cannot occur at the same time. Their intersection is an empty set (A ∩ B = Ø).
Rule: P(A or B) = P(A ∪ B) = P(A) + P(B)

Example: When rolling a die, getting an even number (E = {2, 4, 6}) and getting an odd number (O = {1, 3, 5}) are mutually exclusive.
Independent Events: Two events A and B are independent if the occurrence of one does not affect the probability of the other occurring.
Rule: P(A and B) = P(A ∩ B) = P(A) × P(B)

Example: Tossing a coin twice. The outcome of the first toss does not affect the outcome of the second toss. P(Head on 1st toss and Head on 2nd toss) = P(H) × P(H) = 0.5 × 0.5 = 0.25.
Conditional Events (Conditional Probability): The probability of an event A occurring given that another event B has already occurred.
Rule: P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0.

Example: From a deck of 52 cards, what is the probability of drawing a King, given that the card drawn is a face card? (There are 4 Kings and 12 face cards in total).

Step 1: Identify events. A = drawing a King, B = drawing a face card.

Step 2: Find P(A ∩ B). The intersection is drawing a King (which is also a face card). So, P(A ∩ B) = 4/52.

Step 3: Find P(B). Probability of drawing a face card = 12/52.

Step 4: Apply the formula. P(King | Face Card) = (4/52) / (12/52) = 4/12 = 1/3.

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 asks pupils what they understand by “chance” or “likelihood” in everyday situations, like winning a lottery or predicting weather.
Pupils’ Activity: Pupils share their ideas and examples of situations involving chance.
Learning Point: Pupils connect the concept of chance to the lesson on probability.

Step 2: Meaning of Probability and Classical Approach

Time: 8 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher defines probability and explains the classical approach using a ludo dice. The teacher demonstrates rolling the die and asks pupils about the probability of specific outcomes.
Pupils’ Activity: Pupils observe the demonstration, answer questions, and define probability.
Learning Point: Pupils understand the definition of probability and the classical approach.

Step 3: Frequential Approach

Time: 7 minutes
Teaching Skill: Activity/Explanation
Teacher’s Activity: The teacher leads pupils to perform a simple experiment, such as tossing a coin multiple times, recording outcomes, and calculating the relative frequency of heads or tails. The teacher then explains the frequential approach.
Pupils’ Activity: Pupils participate in the experiment, record data, and calculate relative frequencies.
Learning Point: Pupils understand how probability can be determined from experimental results.

Step 4: Axiomatic Approach, Sample Space, and Event Space

Time: 7 minutes
Teaching Skill: Explanation/Questioning
Teacher’s Activity: The teacher explains the three axioms of probability and defines sample space and event space, guiding pupils to list them for simple experiments like tossing a coin or rolling a die.
Pupils’ Activity: Pupils listen, ask questions, and identify sample spaces and event spaces for given scenarios.
Learning Point: Pupils grasp the fundamental rules of probability and key terminology.

Step 5: Mutually Exclusive Events

Time: 6 minutes
Teaching Skill: Explanation/Example
Teacher’s Activity: The teacher explains mutually exclusive events with examples, such as drawing a red card and drawing a black card from a deck, or rolling an even number and an odd number on a die.
Pupils’ Activity: Pupils identify and explain examples of mutually exclusive events.
Learning Point: Pupils understand mutually exclusive events and their probability rule.

Step 6: Independent Events

Time: 6 minutes
Teaching Skill: Explanation/Example
Teacher’s Activity: The teacher explains independent events using examples like tossing two coins or drawing cards with replacement.
Pupils’ Activity: Pupils listen and work through simple examples of independent events.
Learning Point: Pupils understand independent events and their probability rule.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

Define probability.
State the formula for the classical approach to probability.
List the sample space when two dice are rolled.
Explain the difference between mutually exclusive and independent events with an example for each.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 1 minute
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key concepts of probability, its different approaches, and types of events.
Pupils’ Activity: Pupils listen and make notes.
Learning Point: Pupils recall the main points of the lesson.

Lesson Keywords

Probability – The measure of the likelihood of an event occurring.
Sample Space – The set of all possible outcomes of an experiment.
Event Space – A subset of the sample space, representing specific outcomes.
Mutually Exclusive Events – Events that cannot occur at the same time.
Independent Events – Events where the occurrence of one does not affect the other.
Conditional Events – The probability of an event occurring given that another event has already occurred.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide more complex problems involving combinations of events. For those who need more support, the teacher will provide simplified examples and focused one-on-one guidance during activity time.

Note for teachers using this lesson plan

Ensure that pupils actively participate in the demonstrations and practical examples. Emphasize the real-world applications of probability to make the topic relatable. Encourage pupils to ask questions and discuss their understanding of different event types.