Class: Senior Secondary School 2 (SS2 / SSS 2)
Term: First Term
Week: 7
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Probability
Previous Lesson: Probability: Meaning, Approaches and Types of Events.
Topic: PROBABILITY
Subject Matter: Conditional probability (probability of event given another event has occurred), probability trees

Specific Objectives

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

Cognitive Domain:

Define conditional probability.
State the formula for conditional probability.
Explain the concept of probability trees.

Affective Domain:

Appreciate the application of probability in real-life situations.
Show confidence in solving problems involving conditional probability.

Psychomotor Domain:

Solve problems on conditional probability with and without replacement.
Construct and use probability trees to compute required probabilities.

Work cooperatively to solve probability problems.

Reference Materials

The following resources were used in planning this lesson:

Senior Secondary Schools Education Curriculum
State Unified Scheme of Work
Further Mathematics for Senior Secondary Schools by P.N. Okafor and T.M. Okorie
https://nigerianeducation.gov.ng/
https://waeconline.org.ng/e-learning

Instructional Materials

The teacher will teach this lesson with the aid of:

Ludo dice
Coin
Pack of cards
Whiteboard and markers

Rationale for the Lesson

This lesson enables pupils to understand how the occurrence of one event can affect the probability of another event. It also teaches them a systematic way to visualize and calculate probabilities for sequences of events using probability trees, which is useful in various real-world applications.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of basic probability concepts, including sample space, events, independent events, and the calculation of simple probabilities.

Lesson Content/Board Summary

Conditional Probability and Probability Trees

Conditional Probability

Conditional probability is the probability of an event occurring, given that another event has already occurred. It is denoted as P(A|B), which means the probability of event A happening given that event B has already happened.

The formula for conditional probability is:

P(A|B) = P(A ∩ B) / P(B)

Where:

P(A|B) is the conditional probability of event A given event B.
P(A ∩ B) is the probability of both events A and B occurring.
P(B) is the probability of event B occurring. Note that P(B) must be greater than 0.

Example 1 (Dependent Events – Without Replacement):
A bag contains 5 red balls and 3 blue balls. If two balls are drawn one after the other without replacement, what is the probability that the second ball drawn is red, given that the first ball drawn was red?

Solution:
Let R1 be the event that the first ball drawn is red.
Let R2 be the event that the second ball drawn is red.

Step 1: Find P(R1).
Total balls = 5 + 3 = 8.
P(R1) = Number of red balls / Total balls = 5/8.

Step 2: Find the state of the bag after R1 occurs.
After drawing one red ball, there are 4 red balls left and 3 blue balls left. Total balls = 7.

Step 3: Find P(R2|R1).
P(R2|R1) = Number of remaining red balls / Remaining total balls = 4/7.

Therefore, the probability that the second ball is red given the first was red is 4/7.

Example 2 (Independent Events – With Replacement):
A card is drawn from a standard deck of 52 playing cards, replaced, and then a second card is drawn. What is the probability that the second card drawn is a King, given that the first card drawn was an Ace?

Solution:
Let A1 be the event that the first card drawn is an Ace.
Let K2 be the event that the second card drawn is a King.

Step 1: Understand the effect of replacement.
Since the first card is replaced, the drawing of the first card does not affect the probabilities of the second draw. The events are independent.

Step 2: Apply the property of independent events for conditional probability.
For independent events, P(K2|A1) = P(K2).

Step 3: Calculate P(K2).
There are 4 Kings in a standard deck of 52 cards.
P(K2) = Number of Kings / Total cards = 4/52 = 1/13.

Therefore, the probability that the second card is a King given the first was an Ace (with replacement) is 1/13.

Probability Trees

A probability tree is a diagram used to represent all possible outcomes of a sequence of events and their associated probabilities. Each branch of the tree represents a possible outcome, and the probability of that outcome is written along the branch.

How to construct a probability tree:

Start with a single node representing the beginning of the experiment.
Draw branches from this node for the outcomes of the first event, labeling each branch with its probability.
From the end of each first-event branch, draw further branches for the outcomes of the second event, labeling these with their conditional probabilities.
Continue this process for all subsequent events.
The probability of a sequence of events (a path along the tree) is found by multiplying the probabilities along that path.

Example 3 (Using a Probability Tree – Dependent Events):
A bag contains 4 green (G) marbles and 6 yellow (Y) marbles. Two marbles are drawn without replacement. Draw a probability tree and use it to find the probability of drawing one green and one yellow marble.

Solution:
Total marbles = 4 + 6 = 10.

Step 1: Draw the first set of branches.
P(G1) = 4/10
P(Y1) = 6/10

Step 2: Draw the second set of branches (after drawing one marble without replacement).
If G1 occurred (3G, 6Y left, Total 9):
P(G2|G1) = 3/9
P(Y2|G1) = 6/9

If Y1 occurred (4G, 5Y left, Total 9):
P(G2|Y1) = 4/9
P(Y2|Y1) = 5/9

Step 3: Calculate probabilities of desired paths.
The paths for one green and one yellow marble are (G1, Y2) and (Y1, G2).

P(G1 and Y2) = P(G1) × P(Y2|G1) = (4/10) × (6/9) = 24/90

P(Y1 and G2) = P(Y1) × P(G2|Y1) = (6/10) × (4/9) = 24/90

Step 4: Add the probabilities of the relevant paths.
P(one green and one yellow) = P(G1 and Y2) + P(Y1 and G2)
= 24/90 + 24/90 = 48/90 = 8/15.

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 and asks questions about their previous knowledge of probability, focusing on simple events and independent events.
Pupils’ Activity: Pupils respond to the questions and recall basic probability concepts.
Learning Point: Pupils connect new learning with prior knowledge of probability.

Step 2: Explanation of Conditional Probability

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher defines conditional probability, introduces the formula P(A|B) = P(A ∩ B) / P(B), and explains the terms with simple scenarios like drawing cards without replacement.
Pupils’ Activity: Pupils listen attentively, ask questions for clarity, and copy notes.
Learning Point: Pupils understand the definition and formula for conditional probability.

Step 3: Worked Examples on Conditional Probability

Time: 10 minutes
Teaching Skill: Problem-Solving/Demonstration
Teacher’s Activity: The teacher solves examples on conditional probability, including scenarios with and without replacement, as demonstrated in the board summary. The teacher emphasizes how replacement affects independence.
Pupils’ Activity: Pupils observe the steps, ask questions, and attempt to follow the calculations.
Learning Point: Pupils learn to apply the conditional probability formula to different problem types.

Step 4: Introduction to Probability Trees

Time: 5 minutes
Teaching Skill: Explanation
Teacher’s Activity: The teacher introduces probability trees as a visual tool for analyzing sequences of events, explaining their structure and purpose.
Pupils’ Activity: Pupils listen and understand the concept of a probability tree.
Learning Point: Pupils recognize probability trees as a method for solving complex probability problems.

Step 5: Constructing and Using Probability Trees

Time: 5 minutes
Teaching Skill: Guidance/Demonstration
Teacher’s Activity: The teacher guides pupils through the construction of a simple probability tree on the board, explaining how to label branches with probabilities and how to calculate path probabilities.
Pupils’ Activity: Pupils participate in constructing the tree and understand how to assign probabilities to branches.
Learning Point: Pupils learn the practical steps of drawing and labeling a probability tree.

Step 6: Worked Example using Probability Tree

Time: 5 minutes
Teaching Skill: Problem-Solving/Demonstration
Teacher’s Activity: The teacher solves the example from the board summary involving drawing marbles without replacement, using a probability tree to find the required probability.
Pupils’ Activity: Pupils follow the teacher’s steps and understand how to extract information from the tree to solve problems.
Learning Point: Pupils can use probability trees to solve problems involving multiple 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:

Define conditional probability.
State the formula for conditional probability, P(A|B).
A bag contains 6 red and 4 blue balls. If two balls are drawn without replacement, what is the probability that the second ball is blue, given the first was blue?
Briefly explain when a probability tree is useful.

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 conditional probability and probability trees, emphasizing their importance in solving real-world problems. The teacher then gives pupils homework.
Pupils’ Activity: Pupils listen to the summary and copy down the homework.
Learning Point: Pupils consolidate their understanding of the topic.

Lesson Keywords

Conditional Probability – The probability of an event occurring given that another event has already occurred.
Probability Tree – A diagram showing all possible outcomes of a sequence of events and their probabilities.
Dependent Events – Events where the outcome of one affects the outcome of the other.
Independent Events – Events where the outcome of one does not affect the outcome of the other.
Replacement – Putting an item back after drawing it, affecting subsequent probabilities.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide more complex problems involving three or more sequential events or ask them to create their own probability tree scenarios. For pupils needing more support, the teacher will provide simpler examples with smaller sample spaces and offer one-on-one guidance during problem-solving sessions.

Note for teachers using this lesson plan

Ensure pupils have a solid grasp of basic probability before introducing conditional probability. Emphasize the difference between “with replacement” and “without replacement” as it significantly impacts whether events are dependent or independent. Encourage pupils to draw clear and well-labeled probability trees, as this visual aid is crucial for understanding. Use the instructional materials (dice, cards) to create practical demonstrations and scenarios for pupils to relate to the concepts.