Statistics and
Probability
Quarter 3 – Module 1:
Illustrating a Random Variable
(Discrete and Continuous).
Statistics and Probability – Grade 11
Alternative Delivery Mode
Quarter 3 – Module 1: Illustrating a Random Variable (Discrete and Continuous)
First Edition, 2020
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Statistics and
Probability
Quarter 3 – Module 1:
Illustrating a Random Variable
(Discrete and Continuous)
Introductory Message
For the facilitator:
Welcome to the Statistics and Probability for Senior High School Alternative
Delivery Mode (ADM) Module on Illustrating a Random Variable (Discrete
and Continuous)!
This module was collaboratively designed, developed and reviewed by
educators both from public and private institutions to assist you, the
teacher or facilitator in helping the learners meet the standards set by the K
to 12 Curriculum while overcoming their personal, social, and economic
constraints in schooling.
This learning resource hopes to engage the learners into guided and
independent learning activities at their own pace and time. Furthermore,
this also aims to help learners acquire the needed 21st century skills while
taking into consideration their needs and circumstances.
In addition to the material in the main text, you will also see this box in the
body of the module:
Notes to the Teacher
This contains helpful tips or strategies
that will help you in guiding the
learners.
As a facilitator you are expected to orient the learners on how to use this
module. You also need to keep track of the learners' progress while allowing
them to manage their own learning. Furthermore, you are expected to
encourage and assist the learners as they do the tasks included in the
module.
ii
For the learner:
Welcome to the Statistics and Probability for Senior High School Alternative
Delivery Mode (ADM) Module on Illustrating a Random Variable (Discrete
and Continuous)!
The hand is one of the most symbolized part of the human body. It is often
used to depict skill, action, and purpose. Through our hands, we may learn,
create, and accomplish. Hence, the hand in this learning resource signifies
that you as a learner is capable and empowered to successfully achieve the
relevant competencies and skills at your own pace and time. Your academic
success lies in your own hands!
This module was designed to provide you with fun and meaningful
opportunities for guided and independent learning at your own pace and
time. You will be enabled to process the contents of the learning resource
while being an active learner.
This module has the following parts and corresponding icons:
What I Need to
Know
This will give you an idea of the skills or
competencies you are expected to learn
in the module.
What I Know
This part includes an activity that aims
to check what you already know about
the lesson to take. If you get all the
answers correct (100%), you may decide
to skip this module.
What’s In
This is a brief drill or review to help you
link the current lesson with the previous
one.
What’s New
In this portion, the new lesson will be
introduced to you in various ways such
as a story, a song, a poem, a problem
opener, an activity or a situation.
What is It
This section provides a brief discussion
of the lesson. This aims to help you
discover and understand new concepts
and skills.
What’s More
This comprises activities for independent
practice to solidify your understanding
and skills of the topic. You may check
the answers to the exercises using the
Answer Key at the end of the module.
iii
What I Have
Learned
This includes questions or blank
sentence/paragraph to be filled in to
process what you learned from the
lesson.
What I Can Do
This section provides an activity which
will help you transfer your new
knowledge or skill into real life
situations or concerns.
Assessment
This is a task which aims to evaluate
your level of mastery in achieving the
learning competency.
Additional
Activities
In this portion, another activity will be
given to you to enrich your knowledge or
skill of the lesson learned. This also
tends retention of learned concepts.
Answer Key
This contains answers to all activities in
the module.
At the end of this module you will also find:
References
This is a list of all sources used in
developing this module.
The following are some reminders in using this module:
1. Use the module with care. Do not put unnecessary mark/s on any
part of the module. Use a separate sheet of paper in answering the
exercises.
2. Don’t forget to answer What I Know before moving on to the other
activities included in the module.
3. Read the instruction carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your
answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are through
with it.
If you encounter any difficulty in answering the tasks in this module, do
not hesitate to consult your teacher or facilitator. Always bear in mind
that you are not alone.
We hope that through this material, you will experience meaningful
learning and gain deep understanding of the relevant competencies. You
can
do
it!
iv
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 continues).
1
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.
2. Which of the following is an example of 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. Quantitative variable
B. Discrete variable
C. Qualitative variable
D. Continuous variable
6. This term can best describe a variable that can be counted
A. Continuous
B. Discrete
C. Interval
D. Ratio
2
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. Poison variable
9. Which of the following statement describe a continuous random variable?
A. The average distance traveled by a jeep in a week
B. The number of students present in a Class Anthurium
C. The number of motorcycle 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 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 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 refrigerator 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. Maybe
B. Cannot be determined
C. Yes, it is a random variable
D. No, it is not a random variable
3
14. Which of the following statement 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 belong to 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 19 cases each day
D. The length of time for the check up in the hospital
4
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 outcomes 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 a head or a tail.
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
5
What’s New
Tossing a coin
As you can see in one- peso coin, it has Dr. Jose P. Rizal on one side,
which we will call it as Head (H), and the other side is the Tail (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 when we toss 3 coins?
Continue tossing your coin and record the time. If possible, use
mobile phone timer and record up to the last minutes.
Let say in a minute, how many times the heads and tails appeared.
Then, record all the possible answers on 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 Head, 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 land in eight possible ways:
X = Number of Heads
X
TTT
0
TTH
X
THH
2
1
HTH
2
THT
1
HHT
2
HTT
1
HHH
3
6
Looking at the table we see just 1 case of Three Head, but 3 cases of
Two Heads, 3 cases of One Head, 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 𝑥1 , 𝑥2 , 𝑥3 .....
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
3
7
8
8
8
+ + =
P(X≥1) =1 – P(X < 1) = 1 - P(X = 0)
= 1 – 1/8 = 7/8
7
Example 2;
For a fair coin tosses twice, the probability of each of the possible
values for Number of Heads can be tabulated as shown:
Sample space
Number of Heads
HH
1
HT
2
TH
TT
Number of Heads
3
0
1
2
2/4
Probability
1/4
1/4
or 1/2
Let x is 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, .....
8
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
3
4
4
+ =
P(X≥1) =1 – P(X < 1)
= 1 – 1/4 = 3/4
While 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
measuring.
variable is
a
value
9
that
is
being
acquired
by
What is It
To make you understand better the previous activities, another illustration
is 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
A random variable is a numerical quantity that is assigned to the
outcome of an experiment. We use capital letters to represent a random
variable.
Continuous Data can acquire some value within a range (like for
example a person's height)
10
What’s More
This comprises activities for independent practice to solidify your
understanding and skills of the topic.
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 that point, recognize the two types of arbitrary factors. These are
the discrete and continuous random variables.
Discrete Random Variables are variables can take on a finite
number of distinct values. Examples are 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 to 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.
11
Now, let us try to look to some examples of random variables from the table
below.
Table 1. Examples of Random Variables
Experiment
Number X or the
Random Variable X
Types of Random
Variable
1. Record the number of hours an
specific student use their
mobile from 8:00 am to 5:00
pm for the past three nights
The number of hours
an specific student use
their mobile from 8:00
am to 5:00 pm
Discrete
2. Buying two trays of egg in the
market
The weight of eggs in
kilograms
Continuous
3. Recording of the gender of
family members in a family
with three children
The number of boys
among the children
4. Students will prepare for a quiz
in Mathematics
5. Rolling a pair of dice
How much time
spends reviewing for
this quiz
Discrete
Continuous
Numbers appeared in a
Discrete
pair of dice
What I Have Learned
A. Complete the following statements by writing the correct word.
1. A variable whose value is obtained by counting data is called__________
2. A variable whose value is obtained by measuring is called_____________
3. A variable that can be discrete or continuous is ______________________
4. Time it takes to get to school is an example of ________________________
5. Number of heads in flipping coins is an example of ___________________
12
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 ask 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
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 heights 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
13
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 statement describe a continuous random variable?
A. The number of students present in a Class Temperance
B. The average distance travelled by a tricycle in a month
C. The number of motorcycle owned by a randomly selected household
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
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 book 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 box 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 a section Prudence
C. Number of blue marbles in the box
D. Weight of potatoes in the basket
14
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 refrigerator sell each day
C. Number of late comers in going to school each day
D. Number of people went to the Rizal Park from Monday to Friday
11. Which of the following is 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
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 variable is 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.
15
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. Maybe
B. Cannot be determined
C. Yes, it is a random variable
D. No, it is not a random variable
15. Which of the following statement 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
16
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
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Voter #4
Value of Random
Variables
(Number of Yes votes)
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.C
15.C
18
What I
can do
What I have
learned
A. Complete the
following statements by
writing the correct
word.
1. Discrete
2.Continuous
3.Random Variable
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
1.
2.
3.
4.
5.
C
C
C
D
D
Assessme
nt
1. A
2. B
3. B
4. A
5. D
6. A
7. C
8. A
9. B
10. A
11. A
12. D
13. C
14. C
15. C
Answer Key
19
ADDITIONAL ACTIVITIES
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
Y
1
Voter #2
Voter # 1
Event
Y
Y
Y
N
N
N
N
Y
Y
Y
Y
N
N
N
N
N
N
Y
N
N
Y
Y
Y
N
N
Y
N
N
Y
Y
Y
N
N
Y
N
N
Y
Y
Y
N
N
Y
N
N
Y
Y
Y
Voter #4
Voter #3
Value of Random
Variables
(Number of Yes votes)
4
3
3
2
3
2
2
1
3
2
2
1
2
1
1
0
1 case of four head, 3 cases of Three Heads, 6 cases of Two Heads, 4 cases of One
Head, 1 case of Zero





P (X = 4) = 1/16
P (X = 3 ) = 3/16
P (X = 2 ) = 6/16 or 3/8
P (X = 1 ) = 4/ 16 or ¼
P ( X = 0 ) = 1/16
References
Pierce, Rod. (3 Feb 2020). "Random Variables". Math Is Fun. Retrieved 24 May
2020 from http://www.mathsisfun.com/data/random-variables.html
Malate, Jose S. Statistics and Probability for Senior High School. Vicarisg
Publications and Trading, Inc, 2017.
Lim, Yvette F., Nocon, R., Nocon, E., Ruivivar, L. Math for Engaged Learning
Statistics and Probability. Sibs Publishing House, Inc. 2016.
20
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