Further Mathematics - Senior Secondary 2 - Probability

Probability

Get it on Google Play
Get it on the Apple App Store
  1. Home
  2. Comprehensive Lesson Plan and Notes
  3. Senior Secondary 2
  4. Probability

TERM: 1ST TERM

WEEK 6
Class: Senior Secondary School 2
Age: 16 years
Duration: 40 minutes of 4 periods
Subject: Further Mathematics
Topic: Probability
Focus: Classical, Frequential, and Axiomatic Approaches to Probability, Sample Space, Event Space, Mutually Exclusive, Independent, and Conditional Events
Instructional Resource: Ludo dice, coin, pack of cards

 

SPECIFIC OBJECTIVES:

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

  1. Define and differentiate between the classical, frequential, and axiomatic approaches to probability.
  2. Identify the sample space and event space for different random experiments.
  3. Understand and apply the concepts of mutually exclusive events, independent events, and conditional events.

 

INSTRUCTIONAL TECHNIQUES:

  • Question and answer
  • Guided demonstration
  • Discussion
  • Practice exercises
  • Analogy and real-life connections

 

INSTRUCTIONAL MATERIALS:

  • Ludo dice
  • Coin
  • Pack of cards
  • Whiteboard and markers

 

PERIOD 1 & 2: Introduction to Probability and Classical Approach

PRESENTATION:

Step

Teacher's Activity

Student's Activity

Step 1

Introduces the concept of probability as a measure of uncertainty. Defines probability as a number between 0 and 1.

Students listen attentively and take notes.

Step 2

Introduces the classical approach to probability, where outcomes are equally likely. Uses a coin flip and dice roll as examples.

Students observe examples and participate in questions and answers about the classical probability.

Step 3

Demonstrates probability calculation using the classical approach (e.g., probability of getting heads in a coin flip = 1/2).

Students take notes on how probability is calculated using the classical approach.

Step 4

Links the classical approach to real-life examples, such as rolling a dice or flipping a coin.

Students work in pairs to calculate the probability of different outcomes using coins and dice.

NOTE ON BOARD:

  • Classical Probability:
    • P(E) = Number of favorable outcomes / Total number of outcomes
    • Example: Probability of getting heads in a coin flip = 1/2

EVALUATION (5 exercises):

  1. What is the probability of rolling a 4 on a fair 6-sided die?
  2. What is the probability of getting tails in a coin flip?
  3. Calculate the probability of drawing an Ace from a standard deck of cards.
  4. If a coin is flipped 3 times, what is the probability of getting heads all three times?
  5. If you roll a die, what is the probability of rolling a number greater than 4?

CLASSWORK (5 questions):

  1. What is the probability of drawing a red card from a standard deck of cards?
  2. If you roll a die, what is the probability of rolling a number less than 3?
  3. What is the probability of getting an even number on a 6-sided die?
  4. What is the probability of drawing a king from a deck of 52 cards?
  5. Calculate the probability of getting a sum of 7 when rolling two dice.

ASSIGNMENT (5 tasks):

  1. What is the probability of flipping two coins and both landing heads?
  2. Find the probability of drawing a Queen from a deck of cards.
  3. If you roll a 10-sided die, what is the probability of rolling a 5?
  4. If a dice is rolled twice, what is the probability of getting the same number both times?
  5. What is the probability of drawing a face card (King, Queen, or Jack) from a deck of 52 cards?

 

PERIOD 3 & 4: Frequential and Axiomatic Approaches to Probability

PRESENTATION:

Step

Teacher's Activity

Student's Activity

Step 1

Introduces the frequential approach to probability, where probability is defined as the long-run relative frequency of an event occurring.

Students listen and take notes on the difference between classical and frequential probability.

Step 2

Demonstrates the frequential approach using an experiment: flipping a coin 100 times and recording the frequency of heads.

Students observe and record results in their notebooks during the experiment.

Step 3

Introduces the axiomatic approach to probability, defining it through a set of axioms (rules). Discusses the basic properties of probability.

Students take notes and ask questions for clarification.

Step 4

Links the axiomatic approach to real-life events and examples like coin flips, dice rolls, and card draws.

Students work on exercises using the axiomatic approach to determine the probability of various events.

NOTE ON BOARD:

  • Frequential Approach:
    • Probability of an event is the limit of the relative frequency of the event occurring as the number of trials approaches infinity.
    • Example: Flip a coin 1000 times, the probability of heads is approximately 1/2.
  • Axiomatic Approach:
    • Defined by axioms, such as:
      • P(E) ≥ 0 for any event E.
      • P(S) = 1, where S is the sample space.
      • If E₁ and E₂ are mutually exclusive, P(E₁ ∪ E₂) = P(E₁) + P(E₂).

EVALUATION (5 exercises):

  1. Define the frequential approach to probability.
  2. What is the axiomatic approach to probability?
  3. If a coin is flipped 1000 times, and heads appear 480 times, what is the relative frequency of heads?
  4. What is the probability of drawing a red card from a deck of 52 cards using the axiomatic approach?
  5. Explain the difference between classical and frequential approaches to probability.

CLASSWORK (5 questions):

  1. Use the frequential approach to find the probability of drawing a heart from a deck of cards if 13 hearts are drawn in 100 trials.
  2. What is the probability of drawing a card from a deck that is not a face card, using the axiomatic approach?
  3. Use the axiomatic approach to calculate the probability of an event that is guaranteed to happen.
  4. In an experiment of 500 trials, a coin was flipped 200 times landing on heads. Calculate the relative frequency of heads.
  5. How does the axiomatic approach relate to the concept of a sample space?

ASSIGNMENT (5 tasks):

  1. Research a real-life example of how the frequential approach to probability is applied.
  2. Using the axiomatic approach, calculate the probability of a dice roll resulting in an even number.
  3. Describe how the axiomatic approach can be used to model the probability of rolling a 5 on a 6-sided die.
  4. Explain how the classical approach differs from the axiomatic approach using a coin flip as an example.
  5. Write about an experiment where you could apply the frequential approach to probability.

 

PERIOD 5 : Mutually Exclusive, Independent, and Conditional Events

PRESENTATION:

Step

Teacher's Activity

Student's Activity

Step 1

Introduces the concept of mutually exclusive events, where two events cannot happen at the same time.

Students take notes and discuss examples of mutually exclusive events, like drawing a red card or a black card from a deck.

Step 2

Explains independent events, where the occurrence of one event does not affect the occurrence of another event.

Students work with examples, such as rolling a die and flipping a coin, to identify independent events.

Step 3

Discusses conditional events, where the probability of one event depends on the occurrence of another event.

Students engage in conditional probability problems, such as the probability of drawing a red card given that the first card was a red card.

Step 4

Provides examples and exercises for mutually exclusive, independent, and conditional events.

Students complete exercises and practice examples to reinforce understanding.

NOTE ON BOARD:

  • Mutually Exclusive Events: Events that cannot happen at the same time. Example: Drawing a red card or a black card.
  • Independent Events: The occurrence of one event does not affect the occurrence of another. Example: Flipping a coin and rolling a die.
  • Conditional Events: The probability of one event given that another event has occurred. Example: Probability of drawing a red card given that the first card was red.

EVALUATION (5 exercises):

  1. Give an example of two mutually exclusive events.
  2. Provide an example of two independent events.
  3. Solve for the conditional probability of drawing a red card given that the first card was red.
  4. What is the probability of getting a 3 and then a 4 on two rolls of a die? Are these events independent?
  5. Explain how the occurrence of one event can affect the probability of another in conditional events.

CLASSWORK (5 questions):

  1. Are drawing a king and drawing a queen from a deck of cards mutually exclusive events?
  2. If you flip a coin and roll a die, are these events independent?
  3. What is the conditional probability of getting a head on the second flip of a coin, given that the first flip was tails?
  4. Can you have two independent events that are mutually exclusive? Explain.
  5. Provide an example of a conditional event in a real-life situation.

ASSIGNMENT (5 tasks):

  1. Research and write about a real-world application of mutually exclusive events.
  2. Describe a scenario where you would use conditional probability to make a decision.
  3. Provide an example of two events that are independent in the context of a card game.
  4. Solve a problem involving mutually exclusive events.
  5. Solve a problem involving independent events.