This is the Grade 6 Mathematics Quarter 4 Module 9 self-learning module, Solving Routine and Non-Routine Problems Involving Experimental and Theoretical Probability. It was made by DepEd Region VI for home-based learning. It helps Grade 6 learners use what they know about probability to solve word problems, with a clear four-step plan and many practice items.
What learners will learn
After going through this module, learners are expected to:
- solve routine and non-routine problems involving experimental and theoretical probability.
The module builds on earlier lessons, where learners made simple predictions of events based on experiment results and theoretical probability.
Theoretical and experimental probability
The module uses two formulas.
| Type | Formula | When to use it |
|---|---|---|
| Theoretical probability | number of favorable outcomes ÷ number of possible outcomes | When you know all the possible outcomes, such as the marbles in a bag or the faces of a die |
| Experimental probability | number of times the favorable outcome occurs ÷ number of trials in the experiment | When you have results from actual trials, such as coin tosses or spins |
Answers are written as a simplified fraction and as a percent, for example 1/3 or 33.33%.
The four-step plan
Every problem in the module is solved with the same plan:
- Understand. What is asked? What are the given facts?
- Plan. Which operations will you use? Which formula or equation fits the problem?
- Solve. Use the formula and the given facts, and show the computation.
- Check. Make sure the given values are put into the formula correctly, and check the flow of your computation.
Worked example 1: theoretical probability
A bag has 12 red, 8 blue, and 4 yellow marbles. What is the theoretical probability of getting a blue marble?
- Understand: there are 8 blue marbles out of 24 marbles in all.
- Plan: add to get the total, then divide. TP (blue) = 8/24.
- Solve: divide the numerator and denominator by 8. The answer is 1/3, or 33.33%.
- Check: go back over the values and the computation.
Worked example 2: experimental probability
A coin is tossed 20 times and lands on tails 14 times. What is the experimental probability of heads?
- Understand: 20 tosses, 14 tails.
- Plan: subtract to find the number of heads, then divide. N = 20 − 14.
- Solve: N = 6 heads. EP (head) = 6/20 = 3/10, or 30%.
- Check: review the substitution and the steps.
What the practice activities cover
- What I Know (pretest): six problems about spinners, coffee sales, fish caught, marbles, letters in the word SCHOOL, and raffle tickets.
- What’s In: a review of basic probability with coins, a number cube, numbers 1 to 30, and marbles.
- What’s More: problems with a die, a table of 50 die rolls, and a hands-on activity. Learners draw colored sticks from a box, record the color, return the stick, and repeat at least 10 times.
- What I Can Do: problems about a game show envelope, free throws, a spinner table, a queen from a deck of 52 cards, and cars passing a house.
- Assessment: five problems, including a number from 4 to 10, the letters of LOVE, crayons, a restaurant survey, and neckties.
- Additional Activities: ball-drawing experiments, plus listing all possible outcomes for tossing 3 coins, tossing a coin and rolling a die, and scheduling a weekday morning or afternoon appointment.
If the materials are not available at home, the module suggests using small, equal pieces of colored paper. An ordinary box can also replace a jar.
How to use this module
- Learners: Start with What I Know to check what you already know. Then go through each part in order. Write all your answers on a separate sheet of paper, not on the module. Use the four-step plan for every problem, and check your work against the answer key honestly.
- Parents and facilitators: Help gather simple materials for the experiments, such as sticks, balls, or cut colored paper. Encourage your child to ask the teacher when a task is difficult.
- Teachers: The answer key covers the pretest, What’s In, What’s More, What I Can Do, the Assessment, and the theoretical parts of the Additional Activities. Results of the hands-on experiments will vary from learner to learner.
Frequently asked questions
What grade and quarter is this probability module for?
It is Mathematics Grade 6, Quarter 4, Module 9.
What is the difference between theoretical and experimental probability?
Theoretical probability divides the number of favorable outcomes by the number of possible outcomes. Experimental probability divides the number of times the favorable outcome happened by the number of trials actually done.
What is the four-step plan in this module?
The four steps are Understand, Plan, Solve, and Check. Learners find what is asked and what is given, choose the operations and formula, compute, and then review their work.
Does the module have an answer key?
Yes. It has answers for the pretest, What’s In, What’s More, What I Can Do, the Assessment, and the theoretical parts of the Additional Activities. Experiment results may vary.
What if we do not have marbles or balls at home for the experiments?
The module suggests cutting colored paper into small, equal pieces to use instead. An ordinary box can be used in place of a jar.
Full text of the module
The text below is a transcription of the PDF above, as printed.
Mathematics
Quarter 4 – Module 9 Solaing Routine and Non-Routine Problems Involving Experimental and Theoretical Probability
6
CO_Q4_Mathematics 6_ Module 9
[illustration: students studying outdoors and at a table, a teacher, and a globe, with a lamp]
Mathematics – Grade 6 Alternative Delivery Mode Quarter 4 – Module 9: Solving Routine and Non-Routine Problems Involving Experimental and Theoretical Probability First Edition, 2020
Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties.
Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them.
Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio
Development Team of the Module
Author: Belth Babby P. Anig-ig Editors: Mae Joy M. Tan, Susana L. Lutero, Nora B. Rebadomia, Mary Jane O. Amante, Ma. Pinky C. Arquio, Susan T. Dela Torre Reviewers: Ivy Joy A. Torres, Ma. Theresa L. Tabotabo, Jem Rymon S. Chien, Mae Joy M. Tan Illustrator: Eldiardo E. Dela Peňa Layout Artists: Eldiardo E. Dela Peňa, Rosel P. Patangan Management Team: Ramir B. Uytico, Elena P. Gonzaga, Pedro T. Escobarte, Jr., Donald T. Genine, Adonis A. Mosquera, Clarissa G. Zamora, Fevi S. Fanco, Ivy Joy A. Torres, Jason R. Alpay, Mae Joy M. Tan, Jem Rymon S. Chien, Ethel S. Gali
Printed in the Philippines by_____________________________
Department of Education Region VI
Office Address: Duran Street, Iloilo City Telefax: (033)336-2816, (033)509-7653 E-mail Address: region6@deped.gov.ph
6
Mathematics
Quarter 4 – Module 9: Solving Routine and Non-Routine Problems Involving Experimental and Theoretical Probability
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-by-step 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
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This module was designed and written with you in mind. It is here to help you master the skills in solving routine and non-routine problems involving experimental and theoretical probability. The scope of this module allows you to use it in many different learning situations. The language used recognizes your diverse vocabulary level. The lessons are arranged to follow the standard sequence of your course. But the order in which you read them can be changed to match with the textbook you are now using.
After going through this module, you are expected to solve routine and non-routine problems involving experimental and theoretical probability.
What I Know
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DIRECTIONS: Read and solve each problem. Write your answer in your answer sheet.
- A spinner has 5 sectors namely: red, blue, orange, yellow, and black. If you spin once, what is the probability that the spinner lands on white?
- A coffee shop has sold 20 cups of coffee including 12 brewed coffees. What is the experimental probability that brewed coffee is sold next?
- Aimee together with her friends was on weekend fishing trip. She caught 8 carp, 5 tilapia, and 12 catfish. What is the probability that the next fish which Aimee catches is a tilapia?
- In a jar, there are 7 blue marbles, 3 white marbles, 10 red marbles and 12 marbles. What is the theoretical probability of picking red marbles at random?
- What is the probability of drawing a consonant from the letter of the word SCHOOL?
- There are 500 tickets left in the drop box. What is the probability that a ticket holder will win a prize if she has 20 tickets in all?
Lesson 1: Solving Routine and Non-Routine Problems Involving Experimental and Theoretical Probability
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In the previous lessons, you have learned how to make simple predictions of events based on the results of experiments and theoretical probability. This time, we will focus on how to solve routine and non-routine problems involving experimental and theoretical probability.
What’s In
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Directions: Read and answer the following problems. Write your answer in your answer sheet.
- What is the probability of getting tail when a coin is tossed?
- A fair number cube is rolled. What is the probability of rolling a 4?
- In a cup are the numbers 1 up to 30. Without looking, what is the probability of picking even numbers?
- A bag has 4 red marbles, 3 blue marbles, 2 yellow marbles and 1 green marble. What is the probability of pulling a red marble?
- A fair number cube is rolled five times and these are the results: 2,4,4,5 and 5. Based on the given information, what is the probability that the next roll will be 2?
What’s New
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Read and understand problem below:
A bag contains 12 red marbles, 8 blue marbles, and 4 yellow marbles. Find the theoretical probability of getting a blue marble.
How will you solve the above problem?
What is It
[illustration: magnifying glass with question mark, part-title icon]
The Four-Step Plan can be used in solving the problems.
Let’s solve the first problem.
Step 1. Understand.
a. What is asked? The theoretical probability of getting a blue marble
b. What are the given facts? bag contains 12 red balls, 8 blue marbles, 4 yellow marbles or a total of 24 marbles
Step 2. Plan.
a. What operation to be used to solve the problem? Addition and division are the operations to be used in solving the problem.
b. What formula and equation to be used to solve the problem?
In finding the theoretical probability, use this formula:
Theoretical Probability (event) = $\frac{\text{number of favorable outcomes}}{\text{number of possible outcomes}}$
The equation to be used to solve the problem is:
TP (blue) = $\frac{8}{24}$
[transcriber’s note: fractions printed as stacked numerator over denominator are written with a slash]
Step 3. Solve.
a. Show the computation.
To find the theoretical probability Theoretical Probability (Blue Marble) = number of favorable outcomes / number of possible outcomes
TP(Blue Marble) = 8/24
Simplifying the fraction, 8/24, by dividing both the numerator and denominator by 8 will result to 1/3.
TP(blue marble) = 8/24 ÷ 8/8 = 1/3 or 33.33 %
Therefore, the theoretical probability of getting a blue marble is 1/3 or 33.33 %.
Step 4. Check.
a. Check your answer.
This is one way on how to check if the answer is correct. Go back to your computation. Check if all the given values are properly substituted to the formula. Check also the flow of your computation.
Let’s solve another problem.
A coin is tossed 20 times. It lands on tail 14 times. What is the experimental probability of the coin landing on heads?
Step 1. Understand.
a. What is asked? The experimental probability of the coin landing on heads
b. What are the given facts? a coin tossed 20 times, coin lands on tail 14 times
Step 2. Plan.
a. What operations to be used to solve the problem? Subtraction and Division are the operations to be used in solving the problem.
b. What formula and equation to be used to solve the problem?
- Let N = the number of times that a coin lands on head
N = Total number of times the coin was tossed – the number of times the coin landed on tail
20 times – 14 times = N
- In finding the experimental probability, use this formula:
Experimental Probability (event) = number of times favorable outcomes occur / number of trials in the experiment
EP (Head) = N/20
Step 3. Solve.
a. Show the computation.
- To solve the number of times the coin lands on head (N), do this.
20 times – 14 times = N
6 times = N
Therefore, 6 times the coin lands on head.
- To solve the experimental probability of the coin landing on heads.
Experimental Probability (event) = number of times favorable outcomes occur / number of trials in the experiment
EP (Head) = 6/20
EP (Head) = (6÷2)/(20÷2) = 3/10 or 30%
Therefore, the experimental probability of the coin landing on heads is 3/10 or 30%.
Step 4. Check.
a. Check your answer.
This is one way on how to check if the answer is correct. Go back to your computation. Check if all the given values are properly substituted to the formula. Check also the flow of your computation.
What’s More
Read and solve the problems below. Write your answer in your answer sheet.
When a die is rolled, what is the theoretical probability that number five will be rolled?
Ruby tossed a die 50 times. The results are recorded in the table below. What is the experimental probability that 5 will appear?
| Die Number | Times It Appeared |
|---|---|
| 1 | 10 |
| 2 | 6 |
| 3 | 14 |
| 4 | 6 |
| 5 | 12 |
| 6 | 2 |
A box contains 5 blue sticks, 4 red sticks, and 3 orange sticks. Find the theoretical and experimental probabilities of getting orange stick.
a. Theoretical Probability
b. Experimental Probability
Note: To get the experimental probability:
(1) Take a stick from the bag.
(2) Record the color and return the stick in the bag.
(3) Repeat the process (at least 10 times).
(4) Count the number of times an orange stick is picked (suppose it is 2).
Note: If the materials mentioned are not available at home, colored papers (blue, red and orange) cut into smaller equal pieces can be used for the activity.
What I Have Learned
To solve routine and non-routine problems involving experimental and theoretical probability, follow the 4-step plan:
Step 1: Understand.
a. What is asked in the problem?
b. What are the given facts?
Step 2: Plan.
a. What operations to be used to solve the problem?
b. What formula or equation shall we use to solve the problem?
Step 3: Solve.
a. Solve the problem by using the formula and the given facts.
Step 4: Check.
a. Check if the given values are properly substituted to the formula. Check also the flow of the computation.
What I Can Do
Read and solve the following problems. Write the answer in your answer sheet.
In a popular television game show, a contestant must choose one from the five envelopes. One envelope contains the grand prize, a car. Find the theoretical probability of choosing the envelope that contains the car prize.
James made 15 out of 35 free throws. What is his experimental probability as a fraction?
The table below shows the results of spinning the spinner 100 times. What is the experimental probability of landing on 2?
| Number Spun | Times Spun |
|---|---|
| 1 | 11 |
| 2 | 22 |
| 3 | 15 |
| 4 | 33 |
| 5 | 19 |
There are four queens in a standard deck of 52 playing cards. If Ryle picks a card at random, what is the probability that he picks a queen card?
Kea has seen 12 blue, 15 white and 23 red cars driving by her house in 2 hours. What is the probability that the next car driving by her house is not a white car?
Assessment
DIRECTIONS: Read and answer the following problems. Write your answer in your answer sheet.
You ask a friend to think of a number from 4 to 10. Find the theoretical probability of choosing number 7.
Each letter in the word LOVE are on separate cards, face down on table. You pick a card. What is the theoretical probability that letter V will be picked?
Alex has a box of crayons. The box contains 25 yellow crayons, 15 green crayons, 10 red crayons, 8 orange crayons and 5 red crayons. If he selects randomly a crayon, which color of the crayon she would most likely to select?
A restaurant conducts menu preference survey to its customers. The results of the survey are the following: 8 hamburgers. 15 pizza, 12 pastas, 6 steaks and 9 adobo. What is the experimental probability of hamburgers or steaks?
A drawer contains 6 black neckties, 2 white neckties, 4 red neckties, 2 maroon neckties and 2 blue neckties. Find the theoretical and experimental probabilities of getting black necktie.
a. Theoretical Probabilities
b. Experimental Probability
(Supposed a black necktie is picked 4 times after the experiment is conducted with 32 trials.)
Additional Activities
Read and solve the following problems. Write your answer in your answer sheet.
There are 5 red balls and 6 blue balls in a jar. What are the theoretical and experimental probabilities of picking a red ball?
a. Theoretical probability
b. Experimental probability
Note: To get the experimental probability:
(1) Take a ball from the jar.
(2) Record the color and return the ball in the jar.
(3) Repeat the process (at least 8 times).
(4) Count and record the number of times a red ball IS picked.There are 5 white balls, 8 red balls, and 4 green balls in a jar. Find the theoretical and experimental probabilities of choosing A red ball.
a. Theoretical probability
b. Experimental probability
Note: To get the experimental probability:
(1) Take a ball from the jar.
(2) Record the color and return the ball in the jar.
(3) Repeat the process (at least 10 times).
(4) Count and record the number of times a red ball IS picked.Perform an experiment for each event, and them list all the possible outcomes.
a. Tossing 3 coins
b. Tossing a coin and rolling one regular die
c. Scheduling an appointment for a weekday during the morning or afternoon.
Note: If the materials mentioned are not available at home, colored papers cut into smaller equal pieces can be used for the activity. Ordinary box can be used also instead of a jar.
Answer Key
[transcriber’s note: this page is printed upside down; it is transcribed in upright reading order]
What I Know
- 0
- 3/5 or 60%
- 1/5 or 20%
- 5/16 or 31.25%
- 2/3 or about 66.67%
- 1/25 or 4%
What’s In
- 1/2 or 50%
- 1/6 or about 16.67%
- 1/2 or 50%
- 2/5 or 40%
- 1/5 or 20%
What’s More
- 1/6 or about 16.67%
- 6/25 or 24%
- a. 1/4 or 25%
b. 1/6 or about 16.67%
What I Can Do
- 1/5 or 20%
- 3/7
- 11/50 or 22%
- 1/13 or about 7.69%
- 7/10 or 70%
Assessment
- 1/7 or about 14.29%
- 1/4 or 25%
- yellow
- 7/25 or 28%
- a. 3/8 or 37.5%
b. 1/8 or 12.5%
Additional Activities
- a. 5/11 or about 45.45%
b. Answers may vary - a. 8/17 or about 47.06%
b. Answers may vary
References
Most Essential Learning Competencies (MELCs) in Mathematics 6
Perez, M. et.al, 21st Century MATHletes Learner’s Material, (Quezon City: Vibal Group, Inc. . 2016).
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
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