Determining Outcomes of an Experiment by Systematic Listing
Content Standards and Learning Competencies
Content Standards
The learners should have knowledge and understanding of outcomes from experiments.
Performance Standards
By the end of the lesson, the learners are able to gather data from experiments and represent the data in different forms.
Learning Competencies
1. Express outcomes in words and/or symbols and represent outcomes in tables and/or graphs.
2. Solves problems using the outcomes of experiments.
Learning Objectives
By the end of this lesson, students will be able to:
1. Identify Possible Outcomes
Accurately identify all the possible outcomes in an experiment using systematic listing.
2. Solve Listing Problems
Correctly solve problems involving outcomes in an experiment using systematic listing.
3. Apply Experimental Outcomes
Accurately solve problems using the outcomes of experiments.
4. Illustrate Probabilities
Correctly illustrate the probability of simple events.
5. Solve Probability Problems
Accurately solve problems involving the probability of simple events.
Probability Formula
The Probability Formula
The probability of a simple event, denoted by P(E), is calculated using the formula:
P(E) = n(E)/N
where n(E) is the number of times an event will occur, and N is the total number of possible outcomes.
Activating Prior Knowledge
Short Review
Identify the experiment, outcome, sample space, and event in the following situation:
What is the likelihood of the die landing on a number lower than 4 when thrown?
Experiment:
Outcome:
Sample space:
Event:
Essential Questions
1
When do counting and listing become organized and systematic?
2
How does systematic listing differ from using a table or tree diagram?
3
How can we determine the probability of simple events in various word problems?
4
How can we identify the outcomes of a simple event, and how does this help in calculating its probability?
Introduction to Systematic Listing
Definition
Systematic Listing refers to an organized method of arranging information to ensure that no possible outcomes are overlooked.
Example: Two Coins Toss
List all the possible combinations when two coins were tossed:
(Head, Head)
(Head, Tail)
(Tail, Head)
(Tail, Tail)
Introduction to Outcomes and Experiments
Experiment Definition
In probability, an experiment is any process that leads to an outcome or a set of possible outcomes.
Purpose
Determining the outcomes of an experiment helps us analyze the likelihood of different events occurring.
Example
If you flip a coin, the possible outcomes are heads (H) or tails (T).
Methods for Listing Outcomes Systematically
Listing Method (Enumeration)
Writing down all possible outcomes in an organized manner.
Tree Diagram
Useful when there are multiple stages or choices in an experiment.
Table or Grid Method
Useful for experiments with two variables (e.g., rolling two dice).
Each method helps ensure that no possible outcome is missed.
Listing Method (Enumeration)
Definition
This involves writing down all possible outcomes in an organized manner.
Example: Rolling a six-sided die
Possible outcomes: {1, 2, 3, 4, 5, 6}
Example: Tossing two coins
Possible outcomes: {HH, HT, TH, TT}
Tree Diagram Method
A tree diagram helps organize multiple stages or choices in an experiment, branching out each possibility.
Definition
A tree diagram is useful when there are multiple stages or choices in an experiment.
Example Setup
Rolling a die and flipping a coin.
First Stage
List the six possible outcomes from rolling a die: 1, 2, 3, 4, 5, 6
Second Stage
For each outcome, list the two possibilities for the coin: H or T
Total Outcomes
{1H, 1T, 2H, 2T, 3H, 3T, 4H, 4T, 5H, 5T, 6H, 6T}
Table or Grid Method
A table is useful for experiments with two variables (e.g., rolling two dice). This method displays all possible outcomes in an organized format.
Two-Dice Experiment
When rolling two dice, we have 6 possible outcomes for each die, resulting in 36 total possible combinations.
Complete Grid Layout
The grid shows all 36 possible outcomes as ordered pairs (first die, second die) from (1,1) to (6,6).
Applications
Using this method ensures all 36 outcomes (6 × 6) are identified, helping calculate probabilities of specific events like rolling a sum of 7.
Why Use Systematic Listing?
1
Ensures that no outcomes are missed
Systematic listing provides a structured approach to identify all possible outcomes without omission.
2
Helps calculate probabilities accurately
With a complete list of outcomes, probability calculations become more precise and reliable.
3
Makes complex problems easier to understand
Breaking down complex scenarios into organized lists helps visualize and solve probability problems more effectively.
Example Problem
Problem
A bag contains 3 balls labeled A, B, and C. If two balls are drawn one after the other (without replacement), what are the possible outcomes?
Solution using systematic listing
Possible pairs: AB, AC, BA, BC, CA, CB
Total outcomes: 6
Lesson Activity 1
Problem
Empoy has three balls numbered with the same color: 7, 2, and 5. List all the possible combinations of numbers systematically.
Solution Approach
Use systematic listing to identify all possible combinations of the three numbers.
Consider different arrangements and orders of the numbers 7, 2, and 5.
Lesson Activity 2
During your school's sports festival, students are required to participate in one indoor event and one outdoor event. Systematically list the potential combinations.
Indoor: Badminton
Students can select badminton as their indoor sport option.
Indoor: Table Tennis
Table tennis is another indoor sport choice available.
Indoor: Volleyball
Volleyball offers a team-based indoor option.
Indoor: Futsal
Futsal provides an indoor soccer experience.
Students must choose one indoor event and pair it with one outdoor event:
Outdoor: Football
Football is available as an outdoor team sport.
Outdoor: Swimming
Swimming offers an individual outdoor option.
Outdoor: Relay
Relay races provide a team track event.
Outdoor: Long Jump
Long jump is an individual field event option.
With 4 indoor and 4 outdoor events, there are 16 possible combinations a student can choose from for the sports festival.
Synthesis Activity
What You Have Learned
In a one sheet of paper write something you understand about the lesson we discussed today.
Evaluation Questions 1-2
Question 1
What is the main purpose of systematic listing in probability?
A. To randomly guess the possible outcomes of an experiment
B. To ensure that all possible outcomes are listed without repetition or omission
C. To eliminate certain outcomes based on preference
D. To confuse the experiment with unnecessary steps
Question 2
Which of the following is an example of an experiment in probability?
A. Reading a book
B. Rolling a six-sided die
C. Walking in a straight line
D. Drinking water
Evaluation Questions 3-4
Question 3
If you flip a coin three times, how many possible outcomes can be listed systematically?
A. 3
B. 6
C. 8
D. 9
Question 4
Which of the following is NOT a method of systematic listing?
A. Listing method (enumeration)
B. Tree diagram
C. Guessing method
D. Table or grid method
Evaluation Question 5
Question 5
A box contains three different colored balls: red (R), blue (B), and yellow (Y). If two balls are drawn one after the other (without replacement), how many possible outcomes exist?
A. 3
B. 6
C. 9
D. 12
Answer Key with Visual Explanations
Question 1: B
Systematic listing ensures all possible outcomes are listed without repetition or omission.
Question 2: B
Rolling a six-sided die is a valid probability experiment with clear possible outcomes.
Question 3: C
Three coin flips result in 8 possible outcomes (2³ = 8).
Question 4: C
The "guessing method" is not a valid systematic listing technique.
Question 5: B
Drawing two balls from three colors (without replacement) results in 6 possible outcomes: RB, RY, BR, BY, YR, YB.
Additional Activities for Application
1
Task 1
A coin is flipped twice. List all possible outcomes.
2
Task 2
A spinner has four equal sections labeled A, B, C, and D. What are the possible outcomes when the spinner is spun?
3
Task 3
A six-sided die is rolled. What are the possible outcomes?