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Grade 11 • Unit 1: Random Variables and Probability Distributions
LESSON 1.1
Random Variables
Table of Contents
Learning Competency
1
Learning Objectives
1
Essential Questions
1
Prerequisite Skills and Topics
2
Lesson Proper
3
A.
Introduction to the Lesson
3
B.
Discussion
6
C.
Practice & Feedback
9
Performance Assessment
15
Worksheet Answer Key
16
Synthesis
18
Bibliography
18
Grade 11 • Unit 1: Random Variables and Probability Distributions
Unit 1 | Random Variables and Probability Distributions
Lesson 1: Random Variables
Learning Competency
At the end of the lesson, the learners should be able to illustrate a random
variable (discrete and continuous) (M11/12SP-IIIa-1).
Learning Objectives
At the end of the lesson, the learners should be able to do the following:
●
Define a random variable.
●
Represent real-life situations using random variables.
●
Determine the possible values of a random variable.
Essential Questions
At the end of this lesson, the student should be able to answer the following questions:
●
How does a random variable differ from an algebraic variable?
●
What are some real-life situations that can be represented as random variables?
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Grade 11 • Unit 1: Random Variables and Probability Distributions
Prerequisite Skills and Topics
Skills:
●
Finding the outcomes of a random experiment
●
Determining the sample space of a random experiment
Topics:
●
Math 8 Unit 14: Simple Events | Lesson 1: Experiments, Events, Sample Space, and
Outcomes
●
Math 8 Unit 14: Simple Events | Lesson 2: Determining Outcomes of an Experiment
Using Tables and Tree Diagrams
●
Math 10 Unit 13: Permutation | Lesson 1: Fundamental Counting Principle and the
Factorial Notation
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Grade 11 • Unit 1: Random Variables and Probability Distributions
Lesson Proper
A. Introduction to the Lesson
Suggested Warm-up Activities
Choose from any of the following warm-up activities. These warm-up activities should
either stimulate recall of the previous lesson or introduce the lesson and not already
used in the study guide.
Activity 1: OOTD!
This activity will help the students recall how to list the elements of a sample space
using the Fundamental Counting Principle.
Duration: 10 minutes
Materials Needed: pen, paper
Methodology:
1. Present the following problem together with some pictures.
I have to choose a pair of pants and a shirt. I have two pants and three shirts
in my cabinet. I can also decide whether I am going to wear sunglasses or not.
What are my possible outfits of the day?
2. Ask the students to write the possible outfit combinations in their notebook.
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Grade 11 • Unit 1: Random Variables and Probability Distributions
Expected Results:
a. pants 1, shirt 1, with sunglasses
b. pants 1, shirt 1, without sunglasses
c. pants 1, shirt 2, with sunglasses
d. pants 1, shirt 2, without sunglasses
e. pants 1, shirt 3, with sunglasses
f.
pants 1, shirt 3, without sunglasses
g. pants 2, shirt 1, with sunglasses
h. pants 2, shirt 1, without sunglasses
i.
pants 2, shirt 2, with sunglasses
j.
pants 2, shirt 2, without sunglasses
k. pants 2, shirt 3, with sunglasses
l.
pants 2, shirt 3, without sunglasses
Guide Questions:
1. What difficulty did you encounter in listing the possible outfit combinations?
2. What strategy did you use in listing the possible outfit combinations?
Activity 2: Pass the Task!
This activity will help the students recall how to list the elements of the sample space
of a random experiment.
Duration: 10 minutes
Materials Needed: pen, paper
Methodology:
1. Divide the class into four groups.
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Grade 11 • Unit 1: Random Variables and Probability Distributions
2. Write the following tasks in a card.
Task 1: List all possible outcomes of tossing a coin three times.
Task 2: A fair six-sided die is rolled twice. List all the possible outcomes of the
experiment.
Task 3: Determine the sample space of picking three numbers at random
without replacement from numbers 1 to 3.
Task 4: A mother will give birth to a triplet. What are the possible genders of
the triplet?
3. Give each group a task. Ask each group to list the outcomes using set-builder
notation.
4. After 2 minutes, ask each group to exchange tasks clockwise. Repeat the
process until each group has answered all tasks.
Expected Results:
Task 1: 𝑆 = {𝐻𝐻𝐻, 𝐻𝐻𝑇, 𝐻𝑇𝐻, 𝐻𝑇𝑇, 𝑇𝐻𝐻, 𝑇𝐻𝑇, 𝑇𝑇𝐻, 𝑇𝑇𝑇}
Task 2: 𝑆 = {(1,1), (1,2), (1,3), (1,4), … , (6,6)}
Task 3: 𝑆 = {(1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1)}
Task 4: 𝑆 = {𝐵𝐵𝐵, 𝐵𝐵𝐺, 𝐺𝐺𝐵, 𝐺𝐺𝐺}
Guide Questions:
1. Were you able to accomplish all the tasks?
2. What methods did you use in determining the possible outcomes in each
task?
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Grade 11 • Unit 1: Random Variables and Probability Distributions
Teacher’s Notes
To help better gauge students’ readiness for this lesson, you may assign the short test
given in the Test Your Prerequisite Skills section of the corresponding study guide.
B. Discussion
Suggested Instructional Flow
1. Define and Discover
Teacher’s Notes
You may use the Learn about It! slides in the presentation file to discuss the following
key concepts and examples. Make sure to address student questions before jumping
from one concept to another.
In this lesson, the following key terms and concepts will be discussed:
•
Random Experiment – an experiment that can be repeated numerous times
under the same conditions. The results must be independent of one another.
Example:
Tossing a coin is a random experiment.
•
Outcome – the result of a random experiment.
Example:
The possible outcomes of tossing a coin are head and tail.
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Grade 11 • Unit 1: Random Variables and Probability Distributions
•
Sample Space – the set of possible outcomes of a random experiment;
denoted by a capital letter, usually 𝑆.
Example:
The sample space of tossing a coin is 𝑆 = {𝐻, 𝑇}.
•
Random Variable – a function that associates a numerical value to every
outcome of a random experiment; denoted by a capital letter, usually 𝑋.
The domain is the sample space, and the range is some set of real numbers.
Example:
Suppose 𝑋 represents the number of heads that appear in tossing a coin. The
possible values of the random variable 𝑋 are 0 and 1.
2. Develop and Demonstrate
Example 1
Let 𝑋 be a random variable that denotes the result of rolling a die. What are the
possible values of 𝑋?
Solution:
The sample space of rolling a die is 𝑆 = {1, 2, 3, 4, 5, 6}. Thus, the possible values of
𝑋 are 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, and 𝟔.
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Grade 11 • Unit 1: Random Variables and Probability Distributions
Example 2
There are 10 students who took an examination. If 𝑋 denotes the number of students
who passed the examination, what are the possible values of 𝑋?
Solution:
The number of students who passed the examination could range from 0 (no one
passed) to 10 (everyone passed).
Thus, the possible values of 𝑋 are 𝟎, 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, and 𝟏𝟎.
Example 3
A coin is flipped three times. If 𝑋 represents the number of tails of the outcome, what
are the possible values of 𝑋?
Solution:
1. List the possible outcomes of the experiment.
This can be done using a table or tree diagram.
Let 𝐻 represent heads and 𝑇 represent tails.
From the given illustration, the possible outcomes are:
𝑆 = {𝐻𝐻𝐻, 𝐻𝐻𝑇, 𝐻𝑇𝐻, 𝐻𝑇𝑇, 𝑇𝐻𝐻, 𝑇𝐻𝑇, 𝑇𝑇𝐻, 𝑇𝑇𝑇}
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Grade 11 • Unit 1: Random Variables and Probability Distributions
2. Count the number of tails in each outcome.
Possible outcomes
Number of tails (𝑿)
HHH
0
HHT
1
HTH
1
HTT
2
THH
1
THT
2
TTH
2
TTT
3
Based on the table, the number of tails can be 0, 1, 2, or 3.
Thus, the possible values of 𝑋 are 𝟎, 𝟏, 𝟐, and 𝟑.
C. Practice & Feedback
For Individual Practice
1. Ask the students to answer the following problem items individually using pen
and paper.
2. Give students enough time to answer the problem items.
3. Call a random student to show his or her work on the board afterward.
4. Let the student share how he or she comes up with his or her solution.
5. Inform the student the accuracy of his answer and solution, and in the case
when there is some sort of misconception, lead the student to the right
direction to find the correct answer.
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Grade 11 • Unit 1: Random Variables and Probability Distributions
Problem 1
Let 𝑀 be the number of males in a class with thirty students. What are the possible
values of 𝑀?
Solution:
The number of males in a class could range from 0 (all the students in the class are
females) to 30 (all the students in the class are males).
Thus, the possible values of 𝑀 are 𝟎, 𝟏, 𝟐, 𝟑, …, 𝟐𝟗, 𝟑𝟎.
Problem 2
A mother will give birth to a twin. If 𝑋 denotes the number of boys, what are the
possible values of 𝑋?
Solution:
1. List the possible outcomes. Let 𝐵 represent boys and 𝐺 represent girls.
𝑆 = {𝐺𝐺, 𝐵𝐺, 𝐵𝐵}
2. Count the number of boys in each outcome.
Possible outcomes
Number of boys (𝑿)
GG
0
BG
1
BB
2
Based on the table, the number of boys can be 0, 1, or 2.
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Grade 11 • Unit 1: Random Variables and Probability Distributions
Thus, the possible values of 𝑋 are 𝟎, 𝟏, and 𝟐.
Problem 3
A die is rolled twice. Let 𝑌 be a random variable that denotes the number of even
numbers that appear. What are the possible values of 𝑌?
Solution:
1. List the possible outcomes of the experiment.
𝑆 = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), … , (6,6)}
2. Determine the number of even numbers that appear in each outcome.
Since each outcome has two numbers, there can only be at most two even
numbers.
The value of 𝑌 is 0 if there is no even number that appeared in the outcome, such
as (1, 1).
The value of 𝑌 is 1 if there is exactly one even number that appeared in the
outcome, such as (2, 1).
The value of 𝑌 is 2 if there are two even numbers that appeared in the outcome,
such as (2, 2).
Thus, the possible values of 𝑌 are 𝟎, 𝟏, and 𝟐.
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Grade 11 • Unit 1: Random Variables and Probability Distributions
For Group Practice
1. Ask the students to form a minimum of two groups to a maximum of five groups.
2. Each group will answer problem items four and five. These questions are meant
to test students’ higher order thinking skills by working collaboratively with their
peers.
3. Give students enough time to analyze the problem and work on their solution.
4. Ask each group to assign a representative to show their solution on the board and
discuss how they come up with their solution.
5. Inform the student the accuracy of his answer and solution, and in the case when
there is some sort of misconception, give the student opportunity to work with
his/her peers to re-analyze the problem, and then lead them to the right direction
to find the correct answer.
Problem 4
A technician must choose three chips at random from four chips, two of which are
defective. Let 𝑍 represent the number of non-defective chips chosen. What are the
possible values of 𝑍 given that the first chip chosen is defective?
Solution:
1. List the possible outcomes of the experiment. This could be done using a table or
tree diagram. Let 𝐷1 and 𝐷2 be the defective chips and 𝐹1 and 𝐹2 be the nondefective chips.
It is given that the first chip chosen is defective. Thus, we have the following tree
diagram.
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Grade 11 • Unit 1: Random Variables and Probability Distributions
Based on the table, we have the following unique outcomes. Note that the
subscripts do not matter since we are only interested in the number of nondefective chips.
𝑆 = {𝐷𝐷𝐹, 𝐷𝐷𝐹, 𝐷𝐹𝐷, 𝐷𝐹𝐹, 𝐷𝐹𝐷, 𝐷𝐹𝐹, 𝐷𝐷𝐹, 𝐷𝐷𝐹, 𝐷𝐹𝐷, 𝐷𝐹𝐹, 𝐷𝐹𝐹, 𝐷𝐹𝐷}
Let us simplify the sample space by removing the outcomes that are repeated.
𝑆 = {𝐷𝐷𝐹, 𝐷𝐹𝐷, 𝐷𝐹𝐹}
2. Count the number of non-defective chips in each outcome.
Possible outcomes
Number of non-defective chips (𝒁)
DDF
1
DFD
1
DFF
2
Based on the table, 𝑍 can be 1 or 2. Thus, the possible values of 𝑍 are 𝟏 and 𝟐.
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Grade 11 • Unit 1: Random Variables and Probability Distributions
Problem 5
In a bowl containing three red marbles and two blue marbles, two marbles are drawn
at random without replacement. The first marble drawn is red. If 𝑌 represents the
number of blue balls drawn, what are the possible values of 𝑌?
Solution:
1. List the possible outcomes of the experiment.
This can be done using a table or tree diagram. Let 𝐵1 and 𝐵2 be the blue
marbles, and 𝑅1 , 𝑅2 , and 𝑅3 be the red marbles.
Based on the table, we have the following unique outcomes. Note that the
subscripts do not matter since we are only interested in the number of blue
marbles.
𝑆 = {𝑅𝑅, 𝑅𝑅, 𝑅𝐵, 𝑅𝐵, 𝑅𝑅, 𝑅𝑅, 𝑅𝐵, 𝑅𝐵, 𝑅𝑅, 𝑅𝑅, 𝑅𝐵, 𝑅𝐵}
Let us simplify the sample space by removing the outcomes that are repeated.
𝑆 = {𝑅𝑅, 𝑅𝐵}
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Grade 11 • Unit 1: Random Variables and Probability Distributions
2. Count the number of blue marbles in each outcome.
Possible outcomes
Number of blue marbles (𝒀)
RR
0
RB
1
Based on the table, 𝑌 can be 0 or 1.
Thus, the possible values of 𝑌 are 𝟎 and 𝟏.
Performance Assessment
This performance assessment serves as a formative assessment, divided into three sets
based on the student's level of learning. Click on the link provided on the lesson page to
access each worksheet.
●
Worksheet I (for beginners)
●
Worksheet II (for average learners)
●
Worksheet III (for advanced learners)
Teacher’s Notes
For a standard performance assessment regardless of the student’s level of learning,
you may give the problem items provided in the Check Your Understanding section
of the study guide.
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Grade 11 • Unit 1: Random Variables and Probability Distributions
Worksheet Answer Key
Worksheet I
1. 𝑋 = 0, 1, 2, … ,10
2. 𝑍 = 0, 1, 2, … , 200
3. 𝑋 = 0, 1, 2, … , 50
4. 𝑋 = 0, 1, 2, … , 60
5. 𝑋 = 1, 2
6. 𝑋 = 0, 1, 2, … , 16
7. 𝑌 = 0, 1, 2, 3
8. 𝑋 = 0, 1, 2, 3
Worksheet II
1. a. 𝑍 = 0, 1, 2
b. 𝐴 = 0, 1, 2, 3
2. a. 𝑋 = 0, 1, 2, 3
b. 𝑌 = 2, 3, 4
3. a. 𝑌 = 0, 1, 2, 3
b. 𝑍 = 2, 3, 4, … , 12
4. a. 𝑋 = 1, 2, 3
b. 𝑋 = 1, 2, 3
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Grade 11 • Unit 1: Random Variables and Probability Distributions
Worksheet III
1. a. 𝑋 = 0, 1, 2, 3
b. 𝑋 = 1, 2, 3
2. a. 𝑌 = 2, 3, 4, 5, 6, 7, 8
b. 𝑍 = 0, 1, 2, 3, 4, 5
3. a. 𝑋 = 2, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36
b. 𝑌 = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
4. 𝑍 = 0, 1, 2, 3, 4
5. 𝑋 = 0, 1, 2, 3, 4
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Grade 11 • Unit 1: Random Variables and Probability Distributions
Synthesis
Wrap-up
To summarize the lesson, ask students the following questions:
1. What is a random variable?
2. How do you determine the possible values of a random
variable?
Application and
To integrate values and build connection to the real world, ask
Values Integration
students the following questions:
1. In what real-life situations can you apply random variables?
2. Why do you think we need to study about randomness?
Bridge to the Next
To spark interest for the next lesson, ask students the following
Topic
questions:
1. Can we always illustrate randomness using whole
numbers? Why do you think so?
2. What are the different classifications of random variables?
Bibliography
Belecina, Rene R. et al. Statistics and Probability. Quezon City: Rex Book Store Inc., 2017.
Kenton, Will. “Random Variable Definition.” Investopedia. Retrieved 18 June 2019 from
http://bit.ly/2Zm6aO0
Khan, Salman. “Random variables.” Khan Academy. Retrieved 18 June 2019 from
https://bit.ly/2Rnzv6t
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Grade 11 • Unit 1: Random Variables and Probability Distributions
Pierce, Rod. “Random Variables.” Math Is Fun. Retrieved June 18, 2019 from
http://bit.ly/2XeWg3k
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