Further Mathematics - Senior Secondary 2 - Probability

Probability

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TERM: 1ST TERM

WEEK 7
Class: Senior Secondary School 2
Age: 16 years
Duration: 40 minutes of 4 periods
Subject: Further Mathematics
Topic: Probability
Focus: Conditional Probability and Probability Trees

 

SPECIFIC OBJECTIVES:

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

  1. Define conditional probability and understand its formula.
  2. Solve problems involving conditional probability.
  3. Understand and construct probability trees.
  4. Use probability trees to solve complex probability problems.

 

INSTRUCTIONAL TECHNIQUES:

  • Question and answer
    • Teacher-guided demonstration
    • Hands-on practice with ludo dice, coin, and pack of cards
    • Group work and problem-solving
    • Real-life scenarios for application

INSTRUCTIONAL MATERIALS:

  • Ludo dice
    • Coins
    • Pack of cards
    • Whiteboard and markers
    • Probability tree diagrams
    • Worksheets for practice problems

 

PERIOD 1 & 2: Introduction to Conditional Probability

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the concept of probability and the definition of conditional probability: P(A

B) = P(A and B) / P(B), explaining that it represents the probability of event A occurring given that event B has occurred.

Step 2 - Example

Solves a real-life example involving conditional probability, e.g., a deck of cards: What is the probability of drawing a red card, given that the card is a heart?

Students observe the example, take notes, and ask questions.

Step 3 - Discussion

Discusses common misconceptions in conditional probability, emphasizing the importance of understanding the relationship between events.

Students engage in a discussion, sharing their understanding and asking questions.

Step 4 - Practice

Solves a few additional examples with the class using the dice, coin, and cards. E.g., What is the probability of rolling a 5 on a die, given that the number rolled is greater than 3?

Students solve similar problems in pairs, offering assistance when needed.

NOTE ON BOARD:

  • Conditional Probability Formula: P(A|B) = P(A ∩ B) / P(B)
  • Example: P(red card | heart) = P(heart ∩ red card) / P(heart) = 1/13 ÷ 1/4 = 4/13

 

EVALUATION (5 exercises):

  1. What is the probability of drawing a queen, given that the card drawn is a face card?
  2. If two dice are rolled, what is the probability of rolling a sum of 8, given that one of the dice shows a 4?
  3. If a coin is flipped, what is the probability of landing heads, given that it is an even-numbered coin?
  4. What is the conditional probability of drawing a red card, given that the card is a heart from a deck of cards?
  5. What is the probability of selecting a prime number, given that the number is less than 10?

 

CLASSWORK (5 questions):

  1. What is the probability of getting a 3 on a dice roll, given that the roll is even?
  2. What is the probability of selecting an ace from a pack of cards, given that the card selected is a spade?
  3. In a bag containing 4 red, 3 green, and 2 blue marbles, what is the probability of drawing a blue marble, given that the marble drawn is not red?
  4. What is the probability of drawing a king, given that the card drawn is a face card?
  5. What is the probability of rolling a 6 on a die, given that the number rolled is greater than 4?

 

ASSIGNMENT (5 tasks):

  1. Solve the following: What is the probability of rolling an odd number on a die, given that the number rolled is greater than 2?
  2. What is the probability of drawing a black card from a deck of cards, given that the card is a spade?
  3. If a coin is flipped, what is the probability of landing heads, given that the flip results in an even outcome?
  4. From a deck of cards, what is the probability of selecting a queen, given that the card selected is a heart?
  5. If two dice are rolled, what is the probability of rolling a sum of 10, given that one of the dice shows a 6?

 

PERIOD 3 & 4: Introduction to Probability Trees

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the concept of probability trees as a method for visualizing and calculating the probabilities of multiple events.

Students listen and ask clarifying questions.

Step 2 - Example

Demonstrates how to construct a probability tree, using simple examples like tossing a coin twice (HH, HT, TH, TT) and calculating the probability of each outcome.

Students observe the example, take notes, and follow along.

Step 3 - Probability Tree Construction

Constructs a probability tree for a more complex situation, such as rolling a die and drawing a card from a deck.

Students practice constructing their own probability trees in pairs.

Step 4 - Guided Practice

Provides practice problems for students to solve using probability trees, such as rolling two dice and calculating the probabilities of certain sums.

Students work in pairs to solve the problems, using probability trees.

NOTE ON BOARD:

  • Probability Tree Example for Coin Toss:
    1st flip: H = ½, T = ½
    2nd flip: H = ½, T = ½
    Probability of HH = ½ × ½ = ¼
    Probability of HT = ½ × ½ = ¼

 

EVALUATION (5 exercises):

  1. Draw a probability tree for flipping a coin twice. What is the probability of getting two heads?
  2. Draw a probability tree for rolling a die and then selecting a card from a deck. What is the probability of rolling a 3 and drawing a heart?
  3. Using a probability tree, calculate the probability of rolling a sum of 7 when two dice are rolled.
  4. Draw a probability tree for selecting a red marble and then drawing a face card from a deck.
  5. Using a probability tree, find the probability of getting an even number on a die roll, followed by drawing a queen from a deck of cards.

 

CLASSWORK (5 questions):

  1. Construct a probability tree for flipping a coin twice. What is the probability of getting exactly one head?
  2. Using a probability tree, find the probability of rolling a 6 on a die, followed by drawing a black card from a deck of cards.
  3. Draw a probability tree for rolling two dice. What is the probability of rolling a sum of 9?
  4. Using a probability tree, calculate the probability of drawing a red card, followed by drawing a king from a deck of cards.
  5. Construct a probability tree for selecting a green marble, followed by rolling a 5 on a die.

 

ASSIGNMENT (5 tasks):

  1. Draw a probability tree for flipping a coin twice. What is the probability of getting at least one tail?
  2. Use a probability tree to calculate the probability of rolling a sum of 8 when two dice are rolled.
  3. Draw a probability tree for selecting a card from a deck and then rolling a 3 on a die. What is the probability of this sequence?
  4. Construct a probability tree for drawing two red marbles without replacement. What is the probability of drawing two red marbles?
  5. Using a probability tree, find the probability of rolling an even number, followed by selecting a diamond card from a deck.