Statistics and
Probability
Quarter 3 – Module 1:
Random Variables
Statistics and Probability – Senior High School
Quarter 3 – Module 1: Random Variables and Probability Distributions
First Edition, 2020
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Statistics and
Probability
Quarter 3 – Module 1:
Random Variables
Introductory Message
For the facilitator:
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 at home. Furthermore, you are
expected to encourage and assist the learners as they do the tasks included
in the module.
For the learner:
As a learner, you must learn to become responsible of your own
learning. Take time to read, understand, and perform the different activities
in the module.
As you go through the different activities of this module be reminded of
the following:
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 Let Us Try before moving on to the other
activities.
3. Read the instructions 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 done.
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!
ii
Let Us Learn
This chapter deals with descriptive statistics, which focuses on sets of
measurable data that people experience in their daily life encounter.
Primarily, the topics are the key concepts of random variables, possible values
of random variables and their applications. Modern technology opens an
avenue for opportunity and connectivity on the exact information needed in
everyday life through various sources such as electronic communications,
television, newspapers and etc. However, such information or data will be
verified accurately through statistical analysis, thus, making statistics is very
useful in all human endeavors. With this, the lessons provide students better
understanding and appreciation of the lessons that involved higher and
practical applications, and interpretations of the concepts to challenge
students to think deeply and sharpen their critical thinking and reasoning
abilities.
At the end of the lessons, the students are expected to:
1. illustrate a random variable (discrete and continuous) - M11/12P-IIIa-1
2.distinguish between a discrete and a continuous random variable M11/12P-IIIa-2
3.find the possible values of random variables - M11/12P-IIIa-3
Let Us Try
Direction: Read each item carefully.Using your answer sheet, determine
whether each of the following has discrete or continuous random variable.
1. The average grade in General Math among ABM – Grade 11 students
2. The temperatures in a certain city on a particular week
Monday
– 32.10C
Tuesday
- 300C
Wednesday – 31.20C
Thursday - 34.50C
Friday
- 37.80C
Saturday
- 39.90C
Sunday
- 41.50C
3. Number of credit cards issued by a bank
4. The average daily allowance of five (5) Grade 12 students are:
1
Student 1
Student 2
Student 3
Student 4
Student 5
- P50.00
- P25.00
- P100.00
- P75.00
- P125.00
5. Number of students who passed the DOST scholarship exam
6. The expenses of five (5) employees in a domestic trip :
Employee 1 - P25,550.00
Employee 2 - P21,450.00
Employee 3 - P35,360.00
Employee 4 - P19,090.00
Employee 5 – P49,995.00
7. The winning numbers in a mega lotto draw for today:
45
16
24
12
10
20
8. Length of sleeps of a baby in a number of hours in the hospital
9. Ranks of Bar passers
10.Length in life extension of a cancer patient
Let Us Study
Random Variables
A variable is a specific characteristic or attribute with the population
of interests that varies or changes to any individual or things.
Example of variables are:
The height and weight among Grade 1 pupils, the Emotional
Quotient among public school teachers, the religion of four job applicants,
the number of school organizations registered in all private universities, and
the number of students who enrolled in the public schools in Region XI.
There are two types of variables:
a. Categorical or Qualitative Variables
They are variables that are classified to some attributes or categories
like gender, religion, political affiliations, academic courses, blood type, civil
status and classroom advisory.
2
b.
Numerical or Quantitative Variables
They are variables that are classified according to numerical
characteristics like temperature, height, weight, average grade, scores in the
quarterly examinations, daily allowance and average internet consumption.
Numerical or quantitative variables are classified as:
b.1
Discrete Variables are variables whose values are obtained by
counting.
Examples are:
Number of children who received with polio vaccine, the number
of males and females who are officially enrolled in public schools,
and a number of public utility jeepney registered in the Land
Transportation Office.
b.2
Continuous Variables are variables whose values are obtained
by using measurement.
Examples are:
Temperature, age, height, distance, IQ, weight and daily
expenses.
A variable is a random variable whose values are determined by chance
or random event. The random variables are defined over discrete and
continuous variable in a sample space, as discrete random variables and
continuous random variables respectively. Usually, letters in the alphabet
are commonly used to represent random variables.
Example 1. In tossing two coins, one may use capital letter X, to denote a
random variable in the statistical experiment, this can be illustrated in the
possible outcomes as :
X = {HH, TT, TH, HT}
The numbers 0, 1, and 2 are assigned as random quantities determined
by the possible outcomes of the experiment. These are assumed values to
represent the number of heads, for instance, when tossing two coins.
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Event
1
2
3
4
Sample
Space/Outcome (X)
HH
TT
TH
HT
No. of Heads (x)
2
0
1
1
Example 2. A box contains one red ballpen, and one black ballpen. Two pens
are drawn in successions without replacement. Find the possible outcomes
and the values y of the random variable Y, where Y is the number of red
ballpens.
Y = {RR, BB, RB, BR}
Event
1
2
3
4
Sample
Space/Outcome (Y)
RR
RB
BR
BB
No. of red ballpens
(y)
2
1
1
0
Let Us Practice
A. Determine whether each of the following has discrete or continuous
random variable. Write D if dicrete random variable and C if continuous
random variable.
1.
Number of foreigners migrating to the Philippines every year for
the last 5 years
Year – No. of foreigners
2011 – 650,000
2012 – 690,000
2013 – 750,000
2014 – 790,000
2015 – 801,500
2.
3.
John’s height is 165.4 cm.
Number of vaccines prescribed by a physician to 5 clinics
Clinic 1 – 50,000
Clinic 2 – 43,100
Clinic 3 – 14,700
Clinic 4 – 10,345
Clinic 5 – 9, 889
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4.
5.
6.
The boiling point of water is 1000C.
The volume of a soft drink bottle is 1 liter.
The monthly income of five (5) working mothers are:
Mother 1 – P15, 000.00
Mother 2 – P10,912.00
Mother 3 – P8,501.00
Mother 4 – P35,550.00
Mother 5 – P23,333.00
7.
8.
9.
In a driving competition, the first driver received a score of 9.5.
There are 51 students in Grade 12 section Plato.
My father bought three (3) bottles of red wine with the following
costs:
Bottle 1 – P2,550.00
Bottle 2 – P650,00
Bottle 3 – P219.00
10.
My brother Bimbo was 10 pounds when he was born.
Let Us Practice More
A.
Tell whether each observation is discrete or continuous random variable.
Write D for discrete random variable and C for continuous random variable.
1.
2.
The number of students who passed the Statistics examination
The weights of five (5) Grade 11 students in pounds are:
Maria - 110
Ana
- 120
Dan
- 155
Cecil - 125
Leonila - 135
3.
The number of children of 3 families in a certain barangay are:
Family 1
-3
Family 2
-7
Family 3
-5
5
4.
The total family expenses in a business trip for the month of
May 2019:
Family 1 – P25,000.00
Family 2 – P35,500.00
Family 3 – P25,525.00
Family 4 – P23,732.00
Family 5 – P29,999.00
5.
The number of students enrolled in General Mathematics is
500.
The average speed of tricycle is 50 km/hr
Her cellphone number is 09051555767.
The weekly allowance of a student studied in a university
The total number of books issued to public schools by Learning
Resource Department:
School 1
- 2,445
School 2
- 5,213
School 3
- 345
School 4
- 1,523
School 5
- 798
6.
7.
8.
9.
10.
B.
The total number of learners registered in a Learning
Information System
Word Problem: (#1-4)
Two hairpins are drawn in succession without replacement from the
box containing 4 yellow hairpins and 3 blue hairpins. Complete the table to
find the sample space/possible outcomes and the values z of the random
variable Z, where Z is the number of yellow hairpin.
Event
1
2
3
4
Sample
space/outcome (Z)
YY
2)___
3)___
BB
Choose the letter of your correct answer.
1.
a. 0
b. 1
2.
a. YY
b. YB
3.
a. YY
b. YB
4.
a. 0
b. 1
6
c. 2
c. BY
c. BY
c. 2
No. of yellow
hairpin(z)
1)_____
1
1
4)____
d. 3
d. BB
d. BB
d. 3
Let Us Remember
A.
Select the letter of your correct answer.
1.
A __________is a specific characteristic or attribute with the
population of interests that varies or changes to any individual or
things.
a. sample space
b. experiment
c. variable
d. sample
2.
The following are continuous random variables except
a. height
b. weight
c. sizes of the shoes
d. the number of students in a classroom
3.
A variable is a _____________ whose values are determined by
chance or random event.
a. random variable
b. sample space
c. possible outcome
d. statistical experiment
4.
The following are discrete random variables except
a. the number of students who passed the Statistics examination
b. the total daily expenses of Grade 12 student
c. the total number of books received by the students
d. the number of students enrolled in the public school
5.
It is a variable whose values are obtained by mere counting.
a. discrete variable
b. continuous variable
c. qualitative variable
d. quantitative variable
6.
It is the common representation of random variable.
a. All letters in the alphabet b. Vowels only
c. Roman Numerals
d. Hindu-Arabic Numeration System
7.
It is a variable whose values is obtained by measuring.
a. discrete variable
b. continuous variable
c. qualitative variable
d. quantitative variable
8 – 10. Select at least three letters representing discrete random variables
a.
The mean score of students in General Math pre-test is 45.
b.
the content of a particular softdrink bottle is 2.5 ounces
c.
the number of parents who received Social Amelioration
Program
d.
the number of polio vaccines given to Grade I pupils
e.
the number of facemask boxes available in the drugstore
f.
the average height of Grade 10 students is 130 cm
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Let Us Assess
A.
In your separate answer sheet, write T if the statement is true. Write F
if the statement is false and kindly supply the correct answer.
1.
A random variable is a specific characteristic or attribute with the
population of interests that varies or changes to any individual or
things.
2.
All categorical variables are also considered as qualitative variables.
3.
The qualitative variables are variables which are classified to all
attributes or categories.
4.
The variables like color of the eyes, religion, political parties, Senior
High School Strands and grade levels are all considered as qualitative
variables.
5.
The numerical variables are also known as qualitative variables.
6.
The variables such as height, National Achievement Test results
among Grade 11 students, sacks of rice in the market and number of
modules received by Grade 12 students are all considered as
quantitative variables.
7.
The two classifications under numerical or quantitative variables are
discrete and continous variables.
8.
The discrete variables are values obtained by mere counting number.
9.
A probability distributions are variables whose values are determined
by chance or random event.
10.
The number of parents who received the eight (8) modules in a certain
public school, the number of children who did not receive the polio
vaccine in a particular clinic are all considered as discrete variables.
B.
Word Problem. ( #1- 6)
A school security guard had returned 3 cell phones to 3 students who
claimed that they lost their cell phones with the same brand and model. If
Mary, Alex and Dref, in that order, received one of the 3 cell phones, complete
the table and list the sample points for the possible orders of returning the
cell phones and find the e of the random variable E that represents the
correct matches.
8
Event
1
2
3
4
5
6
Sample
space/outcome (E)
MAD
MDA
ADM
1.)A____
3.)D___
5.)D____
Select the letter of your correct answer.
1.
a. AMD
b. ADM
2.
a. 0
b. 1
3.
a. DMA
b. DAM
4.
a. 0
b. 1
5.
a. DMA
b. DAM
6.
a. 0
b. 1
c. AAD
c. 2
c. DAA
c. 2
c. DAA
c. 2
No. Sample
points(e)
3
1
0
2.)____
4.)____
6.)____
d. AMM
d. 3
d. DMM
d. 3
d. DMM
d. 3
Let Us Enhance
A.
Fill-in the blanks. Write your answers on another sheet of paper.
1.
A variable is a specific characteristic or attribute with the __________ of
interests that varies or changes to any individual or things.
2.
Categorical variable is to qualitative variable as numerical variable is
to
_____________ variable.
3.
A variable is a __________________whose values are determined by
chance or random event.
4.
_________________are variables that are classified to some attributes of
persons or things.
5.
Usually randoms variables
____________________.
are
9
commonly
represented
by
B.
Select which of the following random variables is either discrete or
continuous. Inside the box, kindly write ONLY the item number which
corresponds to your answer.
1.
2.
3.
4.
5.
6.
7.
8.
the total amount of expenses spent by a mother in a week
the number of automobile accidents each year in Davao Region
the length of time to connect to facebook page in a day
the amount of banana production every month in a farm
the amount of milk in pounds produced yearly by a particular
goat
the number of building permits issued each month by Engineering
Department in a particular city
the amount of time consumed by the student in researching to
a particular subject
The age in years of 8 Covid-19 patients admitted at XYZ private
hospital are as follows:
Angel – 12
Cindy – 15
Alexia - 2
Cherry – 20
Wenny – 25
Amor - 36
Glen - 45
Dref
- 76
9.
Weights in kilos among six (6) athletes in boxing event:
Dref – 56
King - 45
Gerry - 47
Alex - 52
Noel - 55
Joel - 50
10.
The total area of the farm lot for sale with its costs:
100 square meter – P200,000.00
200 square meter – P250,000.00
300 square meter – P300,000.00
400 square meter – P350,000.00
500 square meter – P400,000.00
Discrete Variables
Continuous Variables
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Let Us Reflect
The data or information gathered in any particular survey will be
classified according to a specific variable under study. Any random variable
either discrete or continuous is regarded as a point to any statistical analysis
in finding the sample space or possible outcomes, as well as the values of any
random variable.
The continuous and discrete random variables are
represented as numerical-valued variables which can be measured or
counted variables in which they are most applied in real life situations.
The key concepts of random and probability distributions will help
students apply the appropriate random variables in solving real life problems
requiring a higher form of decision making, including those related to games
of chance.
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Let us
Practice
more
A.
1. D
2. C
3. D
4. C
5. D
6. C
7. D
8. C
9. D
10. D
B.
1. C
2. B/C
3. B/C
4. A
12
Let us
Practice
1. D
2. C
3. D
4. C
5. C
6. C
7. C
8. D
9.D
10 D
Let Us Try
A
Discrete
9.
Continuous
8.
Discrete
7.
Continuous
6.
Discrete
5.
Continuous
4.
Discrete
3.
Continuous
2.
Continuous
1.
10. Continuous
Answer Key
Let us Enhance
13
Let us Assess
Let us Remember
B
4.
4. T
4.Categorical/quali
tative
A
3.
3. F (Some)
3. Random variable
D
2.
2. T
2. Quantitative
C
1.
1. F (Variable)
1. Population
A.
A.
C
8.
8. T
Discrete Variables:
B
7.
7. F (Continuous)
B.
A
6.
6. T
5. Letters in the
alphabet
A
5.
5. F (Quantitative)
2, 4, 6, 8
9. F (Random
variables)
D
9.
10. E
10. T
Continuous
Variables:
B.
1. A
1,3,5,7,9,10
2. B
3. A/B
4. A/B
5. A/B
6. A/B
References
Roland S. Zorilla et.al, Statistics and Probability for Senior High School:
Mutya Publishing House, Inc., 2016, 2 – 13.
Edlyn Castillo-Roc Narag, Basic Statistics with Calculator and Computer
Application: Rex Book Store, Inc., 2010, 16.
Elizabeth B. Parreno, Ronel O. Jimenez, Basic Statistics, A Worktext: C & E
Publishing, Inc., 2006, 3 – 10.
Ronald E. Walpole, Introduction to Statistics Third Edition: EDCA Publishing &
Distribution Corporation 388 Quezon Avenue, Quezon City, 1982, 115-118.
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For inquiries or feedback, please write or call:
Department of Education – Region XI
F. Torres St., Davao City
Telefax: (082) 291-1665, (082) 221-6147
Email Address: lrms.regionxi@deped.gov.ph