Statistics and
Probability
Quarter 3 – Module 1:
Illustrating a Random Variable
(Discrete and Continuous)
CO_Q3_Statistics and Probability SHS
Module 1
Statistics and Probability
Alternative Delivery Mode
Quarter 3 – Module 1: Illustrating a Random Variable (Discrete and Continuous)
First Edition, 2021
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Statistics and
Probability
Quarter 3 – Module 1:
Illustrating a Random Variable
(Discrete and Continuous)
Introductory Message
This Self-Learning Module (SLM) is prepared so that you, our dear learners,
can continue your studies and learn while at home. Activities, questions, directions,
exercises, and discussions are carefully stated for you to understand each lesson.
Each SLM is composed of different parts. Each part shall guide you step-bystep as you discover and understand the lesson prepared for you.
Pre-tests are provided to measure your prior knowledge on lessons in each
SLM. This will tell you if you need to proceed on completing this module or if you
need to ask your facilitator or your teacher’s assistance for better understanding of
the lesson. At the end of each module, you need to answer the post-test to self-check
your learning. Answer keys are provided for each activity and test. We trust that you
will be honest in using these.
In addition to the material in the main text, Notes to the Teacher are also
provided to our facilitators and parents for strategies and reminders on how they can
best help you on your home-based learning.
Please use this module with care. Do not put unnecessary marks on any part
of this SLM. Use a separate sheet of paper in answering the exercises and tests. And
read the instructions carefully before performing each task.
If you have any questions in using this SLM or any difficulty in answering the
tasks in this module, do not hesitate to consult your teacher or facilitator.
Thank you.
What I Need to Know
You have studied probability and frequency distributions in statistics in the
previous level. This module was designed and written collaboratively to help you in
illustrating random variables (discrete and continuous) which are essential in
solving real-life problems.
The scope of this module permits it to be used in many different learning situations.
The language used recognizes the diverse vocabulary level of students. The lesson is
arranged to follow the standard sequence of the course.
The module consists of only one lesson entitled illustrating random variables
(discrete and continuous).
After going through this module, you are expected to:
1. define random variable; and
2. illustrate random variables (discrete and continuous).
What I Know
Before studying this module, take this test to determine what you already know
about the topic covered.
Choose the letter of the best answer. Write the chosen letter on a separate sheet of
paper.
1. Which of the following is NOT a discrete variable?
A. Number of white marbles in the box.
B. Number of students present in the classroom.
C. The weight of a box of soft drinks labeled “8 ounces.”
D. The number of arrivals customers in the restaurants between 7:00 a.m.
to 5:00 p.m.
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2. Which of the following is an example of a discrete variable?
A. Distance travelled between tricycles.
B. Height of the students in a certain class.
C. Number of red marbles in the basket.
D. Weight of student.
3. A variable that can be discrete or continuous is called:
A. Random sample
B. Random notation
C. Random variable
D. Random elimination
4. Which of the following is a variable whose value is obtained by measuring?
A. Continuous
B. Discrete
C. Interval
D. Nominal
5. A variable where the information or data can take infinitely many values is
A. Continuous variable
B. Discrete variable
C. Qualitative variable
D. Quantitative variable
6. This term can best describe a variable that can be counted:
A. Continuous
B. Discrete
C. Interval
D. Ratio
7. A set of numerical values assigned to a sample space is called:
A. Random experiment
B. Random sample
C. Random variable
D. None of the above
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8. A variable whose value could be a finite and countable number is a:
A. Continuous variable
B. Discrete variable
C. Poison variable
D. Qualitative variable
9. Which of the following statements describe a continuous random variable?
A. The average distance traveled by a jeep in a week.
B. The number of students present in Class Anthurium.
C. The number of motorcycles owned by a randomly selected household.
D. The number of girls taller than 5 feet in a random sample of 10 girls.
10. Which of the following is a discrete random variable?
A. Jerwin is 165 cm tall.
B. Jerwin has three sisters.
C. Jerwin weighs 68 kilograms.
D. Jerwin ran 500 meters in two minutes.
11. Which of the following variable is a discrete random variable?
A. The amount of unleaded gasoline in a Suzuki car.
B. The temperature of a cup of coffee served at a restaurant.
C. The number of boys in a randomly selected three-child family.
D. The average amount spent on water bill every month of December by a
randomly selected household in Quezon Province.
12. Which of the following is NOT a discrete random variable?
A. Number of refrigerators sell each day
B. Height of dragon fruit as measured each day.
C. Number of students late in going to school each day.
D. Number of people went to the doctor from Monday to Friday.
13. You decided to conduct a survey of families with five children. You are
interested in counting the number of girls (out of five children) in each family.
Is this a random variable?
A. Yes, it is a random variable.
B. No, it is not a random variable.
C. Maybe, it is a random variable.
D. It cannot be determined.
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14. Which of the following statements DOES NOT describe a continuous random
variable?
A. Height of students in a certain class.
B. The average weight of chicken each day.
C. The number of towns in Quezon Province.
D. The distance travelled by a delivery van in an hour.
15. Which of the following is NOT a continuous random variable?
A. The height of the airplane’s flight.
B. The amount of liquid on a container.
C. The number of COVID-9 cases each day.
D. The length of time for the check up in the hospital.
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Lesson
1
Illustrating a Random Variable
(Discrete and Continuous)
This module will assist you with understanding the way toward illustrating random
variables (discrete and continuous). Let’s proceed and appreciate learning.
What’s In
In the study of basic probability, you have discovered that an experiment is any
movement that should be possible more than once under comparative condition. The
arrangement of every possible outcome of an experiment is what we called a sample
space. You have additionally figured out how to mathematically list down the
conceivable outcome of a given experiment. In tossing a coin, for example, the
potential results are turning up heads or tails.
For you to begin, let us all understand that probability distributions can be
illustrated or classified as discrete probability distributions or as continuous
probability distributions, depending on whether they define probabilities associated
with discrete variables and continuous variables.
A variable X whose value depends on the outcome of a random process is called a
random variable. A random variable is a variable whose value is a numerical outcome
of a random phenomenon.
A random variable is denoted with a capital letter. The probability distribution of a
random variable X tells what the possible values of X are and how probabilities are
assigned to those values.
A random variable can be discrete or continuous.
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What’s New
Tossing a coin
As you can see in a one- peso coin, it has Dr. Jose P. Rizal on one side, which we will
call it as heads (H), and the other side is the tails (T). Toss your one-peso coin three
times and record in your notebook the results of the three tosses. In order to write
the result easily, use letter H for the heads and letter T for the tails.
If the results of your three tosses are heads, tails, heads, then you will write HTH on
your notebook.
Example 1: How many heads appeared when we tossed the one-peso coin three
times?
Continue tossing your coin and record the time. If possible, use mobile phone timer
and record up to the last minute.
Let say in a minute, how many times the heads and tails appeared? Then, record all
the possible answers in your notebook.
Write all eight possible outcomes. You can do this systematically so that you do not
get confused later on.
In this instance, there might be 0 heads, 1 heads, 2 heads or 3 heads.
Thus, the sample space is equal to 0, 1, 2, 3.
Then this time, the results or outcomes are NOT entirely equally likely.
The three coins could land in eight possible ways:
X = Number of Heads
X
X
TTT
0
THH
2
TTH
1
HTH
2
THT
1
HHT
2
HTT
1
HHH
3
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Looking at the table we see just 1 case of “three head,” but 3 cases of “two heads,” 3
cases of “one heads,” and 1 case of “zero heads.” So:
•
•
•
•
P(X=3)
P(X=2)
P(X=1)
P(X=0)
=
=
=
=
1/8
3/8
3/8
1/8
That particular example is a discrete variable. A discrete variable is a variable, which
can only view a countable amount of values. Thus, a discrete random variable X has
possible values 𝑥𝑥1 , 𝑥𝑥2 , 𝑥𝑥3 .....
In graphical form:
3/8
1/2
PROBABLITY
Probability
2/8
1/4
1/4
1/8
0
1
2
VALUE
We can use the probability distribution to answer questions about variable x. In
symbols, we want to find P (X ≥1). We could add probabilities to find the answer:
P(X≥1) = P(X=1) + P(X=2) +P(X=3)
1
8
3
8
3
8
= + + =
7
8
P(X≥1) =1 – P(X < 1) = 1 - P(X = 0)
= 1 – 1/8 = 7/8
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Example 2;
The probability of each of the possible values for number of heads can be
tabulated for a fair coin tosses twice, as shown:
Sample space
Number of Heads
HH
2
HT
1
TH
TT
0
Number of Heads
0
Probability
1/4
1
2/4
or 1/2
2
1/4
Let x be equal to the number of heads observed. x is what we called random variable.
•
•
•
P( X=2) = 1/4
P( X=1) = 2/4
P( X=0) = 1/4
This is again an example of a discrete variable. Thus, a discrete random variable X
has possible values x1, x2 , x3, .....
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In graphical form:
2/4
PROBABLITY
Probability
2/4
1/4
1/4
1/4
0
1
2
VALUE
We can use the probability distribution to answer questions about variable x. In
symbols, we want to find P(X ≥1). We could add probabilities to find the answer.
P(X≥1) = P(X=1) + P(X=2)
1
4
2
4
= + =
3
4
P(X≥1) =1 – P(X < 1)
= 1 – 1/4 = 3/4
Meanwhile, to understand the concept of continuous variable, below are the
examples:
 height of students in class;
 weight of 10 statistics books;
 Time it takes to get to school;
 distance travelled between classes.
A continuous variable is a value that is being acquired by measuring.
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What is It
A Random Variable is a capacity that connects a real number with every component
in the sample space. It is a variable whose qualities are controlled by chance. In this
manner, a random variable is a numerical amount that is derived from the results of
an arbitrary trial or experiment. The word “random” is used often in everyday life.
Types of Random Variables
At this point, we can now recognize the two types of arbitrary factors. These are the
discrete and continuous random variables.
Discrete random variables are variables which can take on a finite number of
distinct values. Examples are the number of heads acquired while flipping a coin
three times, the number of kin an individual has, the number of students present in
a study hall at a given time, and so forth.
You can change the experiment by just flipping a coin twice to make things simpler.
Here, the outcomes will be only four: HH, HT, TH, and TT. In addition, the possible
values of X are 0, 1, and 2.
Continuous Random Variables, then again, are random variables that take an
interminably uncountable number of potential values, regularly measurable
amounts. Examples are the height or weight of an individual, the time an individual
takes for an individual to wash, time, temperature, item thickness, length, age, etc.
For you to better understand the previous activities, another illustration and
examples are shown below.
1. How many outcomes are there in tossing 2 coins? 3 coins? 4 coins?
EVENT
SAMPLE SPACE
2 coins are tossed
HH, HT, TH, TT
3 coins are tossed
HHH, HHT, THH, THT HTH, HTT, TTH, TTT
HHHH, HHTH, HHTT, HHHT, HTHH, HTHT,
4 coins are tossed
THTT, TTHH, HTTH, HTTT, THHH, THHT,
TTTT, THTH, TTHT, TTTH
This illustration shows a discrete variable.
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2. Number of pages in a book is a discrete variable.
3. Time taken to run a race is a continuous variable.
4. Number of matches in a box is a discrete variable.
5. Top speed of a boat is a continuous variable.
What’s More
This comprises activities for independent practice to solidify your understanding and
skills of the topic.
Now, answer the following activities below.
A. Classify each set of data as discrete or continuous.
1. The height of eggplants.
2. The time it takes the mobile phone to die.
3. The number of students in class.
4. Volleyball scores.
5. The number of cars in the parking area.
B. Write letter D if the statement is discrete and letter C if it is continuous
variable.
_____1. A container of water.
_____2. The height of tomato plants.
_____3. Molecules of soft drinks.
_____4. The volume of sphere.
_____5. The weight of bags of mango.
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C. Complete the third column by identifying the type of random variable for
each of the given experiment.
Experiment
Number X or the
Random Variable X
1. Recording the number of
hours a specific student use
his/her mobile from 8:00 am
to 5:00 pm for the past three
nights.
The number of hours a
specific student uses
his/her mobile phone
from 8:00 am to 5:00
pm.
2. Buying two trays of eggs in
the market.
The weight of eggs in
kilograms.
3. Recording of the gender in a
family with three children.
The number of boys
among the children.
4. Preparing for a quiz in
Mathematics.
The time a student
spends in reviewing for
this quiz.
5. Rolling a pair of dice.
The numbers appeared
in a pair of dice.
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Type of Random
Variable
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What I Have Learned
A. Complete the following statements by writing the correct word.
1. A variable that can be discrete or continuous is ____________________________.
2. A variable whose value is obtained by counting data is called _______________.
3. A variable whose value is obtained by measuring is called __________________.
4. Time it takes to get to school is an example of ______________________________.
5. Number of heads in flipping coins is an example of _________________________.
B. Complete the table below.
Experiment
Number X or the
Random Variable X
Types of Random
Variable
1. Number of rings before the
phone is answered.
2. Teacher asking the students
to finish the test after an
hour.
3. Number of complaints per
day.
4. Height of the tallest building
in Lucena City.
5. Number of mobile phones in a
household.
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What I Can Do
Things to do:
Answer the following.
Classify whether the given experiment implies a discrete random variable or a
continuous random variable. Write D if discrete and C if continuous.
_____ 1. The temperature of a solution in the laboratory.
_____ 2. Collecting data about the height of students in a public school.
_____ 3. Recording the distance travelled by the bus.
_____ 4. Surveying about the number of cases due to COVID-19 pandemic
in Quezon Province.
_____ 5. Number of promoted students at the end of school year.
Assessment
Multiple Choice. Choose the letter of the best answer. Write your chosen letter on a
separate sheet of paper.
1. A variable where the information or data can take infinitely many values is:
A. Continuous variable
B. Discrete variable
C. Quantitative
D. Qualitative variable
2. Which of the following statements describes a continuous random variable?
A. The number of students present in section Temperance.
B. The average distance travelled by a tricycle in a month.
C. The number of motorcycles owned by randomly selected households.
D. The number of girls taller than 5 feet in a random sample of 6 girls.
3. A variable that can be discrete or continuous is called:
A. Random sample
B. Random variable
C. Random notation
D. Random elimination
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4. Which of the following is a variable whose value is obtained by measuring?
A. Continuous
B. Discrete
C. Interval
D. Normal
5. Which of the following is NOT a discrete variable?
A. Number of books per student.
B. Number of green marbles in the box.
C. The number of arrivals of customers in the clinic between 8:00 a.m. to
4:00 p.m.
D. The weight of a case of soft drinks labeled 12 ounces.
6. Which of the following is an example of discrete variable?
A. Distance travelled between cars.
B. Height of the students in section Prudence.
C. Number of blue marbles in the box.
D. Weight of potatoes in the basket.
7. A set of numerical values assigned to a sample space is called:
A. Random experiment
B. Random sample
C. Random variable
D. None of the above
8. A variable whose value could be a finite and countable number is a:
A. Continuous variable
B. Discrete variable
C. Qualitative variable
D. Quantitative variable
9. This term can best describe a variable that can be counted.
A. Continuous
B. Discrete
C. Interval
D. Ratio
10. Which of the following is NOT a discrete random variable?
A. Height of eggplant as measured each day.
B. Number of refrigerators sold each day.
C. Number of late comers in going to school each day.
D. Number of people who went to the Rizal Park from Monday to Friday.
11. Which of the following is a discrete random variable?
A. Jose has four sisters.
B. Jose is 163 cm tall.
C. Jose weighs 68 kilograms.
D. Jose ran 300 meters in one and a half minutes .
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12. Which of the following is NOT a continuous random variable?
A. The height of the airplane’s flight.
B. The amount of liquid on a container.
C. The length of time for the check up in the hospital.
D. The number of clients of a certain insurance company each day.
13. Which of the following variables is a discrete random variable?
A. The amount of unleaded gasoline in a Suzuki car.
B. The temperature of a cup of coffee served at a coffee shop.
C. The number of boys in a randomly selected two-child family.
D. The average amount spent on electric bill every month of May by a
randomly selected household in Quezon Province.
14. You decided to conduct a survey of families with three children. You are
interested in counting the number of girl in each family. Is this a random
variable?
A. Yes, it is a random variable.
B. No, it is not a random variable.
C. Maybe, it is a random variable.
D. It cannot be determined.
15. Which of the following statements DOES NOT describe a continuous random
variable?
A. Height of students in a certain class.
B. The average weight of chicken each day.
C. The number of streets at barangay Tahimik.
D. The distance travelled by a delivery van in an hour.
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Additional Activities
Hondagua National High School-Senior High School would like to conduct election
for the Accountancy, Business, and Management (ABM) officers. Complete the table
for the possible outcomes from a sample of four voters and identify also the value of
random variable of the number of “yes” votes.
Event Voter # 1
Voter #2
Voter #3
Voter #4
Value of Random
Variables
(Number of Yes votes)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
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CO_Q3_Statistics and Probability SHS
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What I Know
1. C
2. C
3. C
4. A
5. D
6. B
7. C
8. B
9. A
10. B
11. C
12. B
13. C
14. B
15. C
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What’s More
A. Classify each set of data as discrete or
continuous.
1. Continuous
2. Continuous
3. Discrete
4. Discrete
5. Discrete
B. Write letter D if the statement is discrete and
letter C if it is continuous variable.
1. C
2. C
3. C
4. D
5. D
C. Complete the third column by identifying the
type of random variable for each of the given
experiment.
1. Discrete
2. Continuous
3. Discrete
4. Continuous
5. Discrete
Answer Key
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What I Have Learned
A. Complete the following statements by writing the correct word.
1. Random Variable
2. Discrete
3. Continuous
4. Continuous
5. Discrete
B. Complete the table below.
Random Variable
1. Number of ringing of phone
2. Time of the students to finish the test
3. Number of complaints
4. Height of the tallest building
5. Number of mobile
Types of Random Variable
1. Discrete
2. Continuous
3. Discrete
4. Continuous
5. Discrete
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What I Can Do
20
Assessment
1. C
1. A
2. B
2. C
3. B
3. C
4. A
4. D
5. D
5. D
6. A
7. C
8. A
9. B
10. A
11. A
12. D
13. B
14. C
15. C
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Additional Activities
Complete the table for the possible outcomes from a sample of four voters and
identify the value of random variable of the number of “yes” votes.
N
16
N
15
N
14
N
13
N
12
N
11
N
10
N
9
Y
8
Y
7
Y
6
Y
5
Y
4
Y
3
Y
2
Y
1
Voter # 1
Event
Y
Y
Y
Voter #4
Voter #3
Voter #2
Y
Y
Y
N
N
N
N
Y
Y
Y
Y
N
N
N
N
Y
N
N
Y
Y
N
N
Y
Y
N
N
Y
Y
N
N
Value of Random
Variables
(Number of Yes
votes)
0
N
1
Y
1
N
2
Y
1
N
2
Y
2
N
3
Y
1
N
2
Y
2
N
3
Y
2
N
3
Y
3
N
4
1 case of four yes votes, 3 cases of three yes votes, 6 cases of two yes votes, 4
cases of one yes vote, 1 case of Zero
• P (X = 4) = 1/16
• P (X = 3 ) = 4/16 or 1/4
• P (X = 2 ) = 6/16 or 3/8
• P (X = 1 ) = 4/ 16 or ¼
• P ( X = 0 ) = 1/16
References
Pierce, Rod. (2020). "Random Variables". Math Is Fun. Retrieved 24 May 2020 from
http://www.mathsisfun.com/data/random-variables.html
Malate, Jose S. (2017) Statistics and Probability for Senior High School. Vicarish
Publications and Trading Inc.
Lim, Yvette F., Nocon, R., Nocon, E., Ruivivar, L. (2016) Math for Engaged Learning
Statistics and Probability. Sibs Publishing House Inc.
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For inquiries or feedback, please write or call:
Department of Education - Bureau of Learning Resources (DepEd-BLR)
Ground Floor, Bonifacio Bldg., DepEd Complex
Meralco Avenue, Pasig City, Philippines 1600
Telefax: (632) 8634-1072; 8634-1054; 8631-4985
Email Address: blr.lrqad@deped.gov.ph * blr.lrpd@deped.gov.ph