Probability of a Union Mathematics 25 pag. Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) 10 NOT Mathematics Quarter 3 - Module 7 Probability of a Union of Two Events Department of Education ● Republic of the Philippines Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) Mathematics- Grade 10 Alternative Delivery Mode Quarter 3 - Module 7: Probability of a Union of Two Events 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 royalty. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this book 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, PhD Undersecretary: Diosdado M. San Antonio, PhD Development Team of the Module Author/s: Reviewers: Charmaine P. Yonson Rhodel A. Lamban, PhD Elbert R. Francisco, PhD Samson C. Gaje Merlyn L. Brigoli Illustrator and Layout Artist: Charmaine P. Yonson Management Team Chairperson: Arturo B. Bayocot, PhD, CESO III Regional Director Co-Chairpersons: Victor G. De Gracia Jr., PhD, CESO V Asst. Regional Director Randolph B. Tortola, PhD, CESO IV Schools Division Superintendent Shambaeh A. Usman, PhD Assistant Schools Division Superintendent Mala Epra B. Magnaong, Chief ES-CLMD Neil A. Improgo, EPS-LRMS Bienvenido U. Tagolimot, Jr., PhD, EPS-ADM Members Elbert R. Francisco, PhD, Chief ES-CID Rhodel A. Lamban, PhD, EPS Mathematics Rejynne Mary L. Ruiz, PhD, LRMDS Manager Jeny B. Timbal, PDO II Shella O. Bolasco, Division Librarian II Printed in the Philippines by Department of Education – Division of Bukidnon Office Address: Fortich Street, Sumpong, Malaybalay City Telephone: (088) 813-363 E-mail Address: bukidnon@deped.gov.ph Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) 10 Mathematics Quarter 3 - Module 7 Probability of a Union of Two Events This instructional material was collaboratively developed and reviewed by educators from public and private schools, colleges, and or/universities. We encourage teachers and other education stakeholders to email their feedback, comments, and recommendations to the Department of Education at bukidnon@deped.gov.ph. We value your feedback and recommendations. Department of Education-Division of Bukidnon ● Republic of the Philippines Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) Table of Contents PAGE COVER PAGE COPYRIGHT PAGE TITLE PAGE TABLE OF CONTENTS WHAT THIS MODULE IS ABOUT Note to the Teacher/Facilitator Note to the Learner Note to the Parents/Guardian Module Icons WHAT I NEED TO KNOW 1 WHAT I KNOW(PRETEST) 2 LESSON 1: PROBABILITY OF A UNION OF TWO EVENTS 4 What I Need to Know What I know What’s in What’s New What Is It What’s More What I Have Learned What I Can Do Assessment Additional Activities 4 4 5 5 7 10 11 12 12 13 SUMMARY 14 POSTTEST 15 ANSWER KEY 17 REFERENCES 18 Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) WHAT THIS MODULE IS ABOUT This module will enable the learners to independently provide visual structure of certain given events together with its operations on the union and intersections of these events. The attainment of the main goal of this module was made possible through the appropriate and currently applicable activities formulated by the author to cater the needs of the learners. The necessary details and sufficient discussions needed for the learners to successfully answer all the conditioning exercises were logically provided in this paper so that the relevant concept for this module will be totally mastered. Notes to the Teacher/Facilitator To the teachers, this module is designed to let the learners learn by themselves at times like this crisis that face to face instruction at school cannot be possible. We are responsible to take precautionary measures in ensuring quality education for our learners. It is the ultimate dream in the Department of Education that no learners must be left behind and so realizing this in this unstable situation due to COVID-19 pandemic become even more challenging. But it our oath made that at all times, quality education must be accessible to all no matter how demanding this could be. In this module, you will be teaching the learners the probability of a union of two events. Used the different activities provided here to motivate and challenge the learners to continue learning and be able to adopt the new normal in the field of education nowadays. Notes to the Parents/Guardians To the parents/guardians, this module is for our learners to independently learn the probability of a union of two events. Various tasks are provided here to allow them to fully develop the needed knowledge and skills. Let’s be reminded that education for our learners has never been just the work of the school or teachers but it’s always a combine effort of the school and society. Thus, we must actively participate the mission and vision of the Department of Education especially at this time where every single serious attempt to nurture the learning capacity of our learners is vital. Notes to the Learners To the learners, this module talks about the probability of a union of two events. To be able to understand the topic, this module is made easy for your convenience but the learning you’ll get is sure and compact. The demand of the world to quality education must not be sacrificed at this crisis for if it will happen, it is your future and the future of our nation that will be renounced. The Department of Education has been doing all possible means to deal with the current situation the country has been facing of. Making sure that your education won’t be left behind amidst this pandemic. Administrators, teachers, stakeholders and parents had been working hand in hand to make leaning feasible to the majority. Now, it is in your hand to take part of these efforts and campaign by simply conscientiously reading and answering this module. Do it with honest convection that education is not solely about how teachers provide instructions but it also rely on the commitment of every learner to expand what they have learned, understood what confused them, and acquire information and knowledge new to them. Good luck and God bless learners! Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) Icons of this Module What I Need to Know This will give you an idea of the skills or competencies you are expected to learn in the module. 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 I Know 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. This section provides a brief discussion of the lesson. This aims to help you discover and understand new concepts and skills. What is It 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. What’s More 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. References This contains all the sources which helped in the formulation of the module. Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) What I Need to Know This Quarter 3 - Module 7 is all about the probability of a union of two events. Certain competency and subtask are provided below. Competency: Illustrates the probability of a union of two events. (Code: M10SP-IIIg-1) Subtasks/Objectives: At the end of the week, the students are expected to: 1. define probability; 2. define union of two events; and 3. illustrates the probability of a union of two events. How to Learn from this Module To achieve the objectives cited above, you are to do the following: • Take your time reading the lessons carefully. • Follow the directions and/or instructions in the activities and exercises diligently. • Answer all the given tests and exercises 1 Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) What I Know Direction: Read each item carefully and choose the letter of the correct answer. 1. It is defined as a ratio of how many times an event may occur as compared to the total possible outcomes. A. Chances B. Probability C. Union of events D. Statistics 2. When choosing an “Ace” from a standard deck of cards, the probability is ___. B. C. D. A. 3. The sum of the probability of getting a head and the probability of getting a tail when tossing a coin once is equal to ____. A. 1 B. C. D. 0 4. The school principal decided to repaint the school administration building. The school custodian present four available paint colors to choose from (green, neon pink, yellow and royal blue). What is the probability that the school principal will choose green? B. C. D. A. 5. In an urn, there are 20 balls in which 4 of these are in color blue, 5 are on color orange, 8 are in color violet and 3 are in color red. What is the probability of getting orange when ask to draw one ball from the urn? A. B. C. D. 6. (from # 5) If A is an event of red balls and B is an event of violet balls, find A. B. C. D. 7. What is the probability of picking two spade cards or two face cards? A. B. C. D. 8. In tossing a coin twice, what is the probability of getting one or two heads? C. D. A. 1 B. For numbers 9 to 11, please refer your answers on the figures provided below. Figure 1. 2 Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) 9. What is the probability of event A to happen ( A. B. C. )? 10. What is the probability of event B to happen ( A. B. C. )? 11. Find A. D. D. . B. C. 12. Given two events Y and Z. If the D. and the , then this means that _____. A. the events are not mutually exclusives. . B. C. both events satisfied the condition that D. the events are intersecting at some point. . 13. Given to events A and B. If the and that ______. A. the events are mutually exclusives. B. both events have probability of less than 1. C. the events are intersecting. D. both events satisfied the condition which states that , then this implies . 14. In a Teachers’ Division Seminar for the new normal set-up in school opening via zoom, 90 school heads joined. 53 of the school heads voted for modular class as an alternative way of delivering lessons to the learners (event A) and 70 voted for online class (event B). Find the probability that school heads voted online class only? A. B. C. D. Figure 2. 15. A spinner in figure 2 is equally divided into 8 parts and numbered from 1 to 8. Let and . What is ? A. B. C. D. 0 3 Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) Lesson 1 Probability of a Union of Two Events What I Need to Know This Quarter 3 - Module 7 is all about Probability of a Union of Two Events. In this module you expected to develop competency on the ability to illustrate the probability of a union of two events. To complete this goal, you just have to accomplish certain subtasks. First, you need to define what is meant by the word “probability”. Second, you need to determine union of two events. And lastly, you need to demonstrate the probability of a union of two events. All these three subtasks are provided with appropriate and current activities to entice your interest so you will be able to successfully go through this module with full understanding on the main concept on Probability of a Union of Two Events. What I Know Direction: Identify the correct answer of the following questions. 1. When choosing an “Ace” and a “King” from a standard deck of cards, the probability is __________. Answer: _________________________________________ 2. The sum of the probability obtained in all possible outcomes when throwing a die is equal to ____. Answer: _________________________________________ 3. In an urn, there are 10 balls in which 3 of these are in color blue, 2 are on color orange, and 5 are in color violet. What is the probability of getting orange when ask to draw one ball from the urn? Answer: _________________________________________ 4. (from # 3) If A is an event of blue balls and B is an event of violet balls, find Answer: _________________________________________ 5. What is the probability of picking one heart cards or one face cards? Answer: _________________________________________ 6. In tossing a coin twice, what is the probability of getting two tails? Answer: _________________________________________ 4 Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) 7. In an English alphabet, what is the probability of getting a vowel letter when you’ll be asked to pick one? Answer: _________________________________________ )? When 8. In a word MERCY, what is the probability of event B to happen ( B is an event of obtaining a non-vowel letter. Answer: _________________________________________ 9. Find , when A is the event of the word PRAYERS and B is an event of the word WONDROUS. Answer: _________________________________________ 10. In a set of natural numbers less than 10, when you’ll be asked to pick one, what is the probability of getting a prime number? Answer: _________________________________________ What’s In In the previous module, you have been taught of illustrating events, together with its types and its basic operation on union and intersection. You were able to define event and were also able to identify what type of event applied in a scenario given. Items on finding the union and intersection of events were also given which surely you had successfully answered making you able to proceed in this next module. Those items will be very useful to easily understand what this module all about. After being able to know the union of say event A and B, your next task would be to know the probability that such union would happen. Be ready to learn and enjoy as you go along with this module. What’s New Let’s explore For the 40th day of enhanced community quarantine at Barangay Santa Rita due to the COVID-19 pandemic, Athena tried to prepare breakfast. But she is having hard time in choosing what flavor of can goods is she going to open for meal. If you can still perfectly remember, you helped her choose her viand of the day. You now know that what Athena picked and cooked as viand of the day is an example of an event. Now, lets’ do this. • What is the probability that each can will be chosen? • How will the answer be obtained? 5 Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) • How about the probability of Athena’s viand of the day be any can goods or a hotdog? • Is there a chance that Athena would choose an egg or a canton noodles? Let’s try this Identify the correct answer in each of the questions base on the figures provided. For numbers 1 to 5. 1. Describe event A. …………………………………………… 2. Describe event B. …………………………………………… ? Explain your answer 3. Is shortly. …………………………………………… ? How about ? 4. What is …………………………………………… 5. Find . …………………………………………… For numbers 6 to 10. 6. Describe event X. …………………………………………… 7. Describe event Y. …………………………………………… 8. Is ? Explain your answer shortly. …………………………………………… ? How about ? 9. What is …………………………………………… 10. Find . …………………………………………… 6 Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) For numbers 11 to 15. 11. Describe U. …………………………………………… 12. Describe event M and N. …………………………………………… 13. Is ? Explain your answer shortly. …………………………………………… ? How about ? 14. What is …………………………………………… 15. Find . …………………………………………… What Is It Probability It is defined as a ratio of Example 1: how many times an event In an experiment of may occur as compared to tossing a coin twice, what is the total possible outcomes. the probability that a tail Probability of an event would come out? P(M) is equal to the total Solution: number of desired event (M) Let P(M) be the divided by the total number of probability of getting a tail, possible outcomes (L). M be total number of desired event, and L be the total number of possible outcomes. Note: M = 1 (in tossing a coin ✓ once, only 1 tail is the outcome) ✓ The sum of the probabilities of all L=2 (head and tail) outcomes is equal to 1. (substitute the (Example: In flipping a die once, the possible outcomes are 6. Thus, each outcome has a probability of . Adding all these probabilities ) would result ( to 1.) obtain values) or 0.5 If the answer has to be converted into percentage form, then 0.5 has to be multiplied by 100. 7 Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) Thus, or 50% is the probability of getting a tail in an experiment of tossing a coin. Example 2: What is the probability of getting a 5 when rolling a regular die once? Solution: Let P(X) - be the probability of getting a 5 when rolling a die once. X - be the total number of desired event. Z - be the total number of possible outcomes. X=1 Z=6 (since in a regular die, only one face has a 5 dots mark) (since in a regular die has 6 faces) Thus, or 16.67% is the probability of getting 5 when rolling a regular die once. 4 Union of two events Given two events M and Example: N, the union of these two If given that events events is the elements which & , are found in M, in N or in then Both M and N. . In symbol, . To illustrate. . . Note: The word or here means one or the other, or both. Take note that in listing the elements in the union of two events, the common elements have to be listed once only. Probability of a Union of two Events Given two events M and Example 1: N, the probability of a union In the previous of these two events (which example we have can be written as ) & is equal to the sum of the . probability of event M and the Find . 8 Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) probability of event N minus the probability of (the elements occurring to both events). In this case it is assume that M and N are not mutually exclusive events, in . other words Solution: Event M has 6 elements; event N has 5 elements: and the total number of elements is 11. In symbol, . Note: If these two events (M and N) are mutually exclusive, then . an event of getting head and M is an event of getting tail. Here, there is no chance that the outcome will be of both event making . And so, empty, thus . , one head out of 2 possible outcome one tail out of 2 possible outcome ) , So, (Example: Tossing a coin once, L is . At this point, we have to take note that , meaning there are elements common to both events (as illustrated in the figure above). Now, how many elements found in ? and so . Substituting the values obtained in the equation below, then Thus, the probability of a union of events M and N is . In symbol, . Example 2: In an experiment of drawing a card from a regular deck of cards, what is the probability of drawing a heart or a face card. Solution: Let M be an event with heart cards and N be an event with face cards. Note that the total number of cards in a regular deck of cards is 52. M = 13 cards (since there are 13 heart cards out of 52) and so, 9 Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) N = 12 cards (since there are 12 face cards out of 52) and so, , meaning the two events are not Note that in this case, mutually exclusives. Why? Elements common to both events. 1 Thus, or 50% is the probability of getting a 2 tail in an experiment of tossing a coin. , making It can be seen from the figure that, Substituting the values obtained , then above Therefore, . in the equation . What’s More Let’s do this Define shortly but concisely the terms listed below and provide simple illustration to help define the terms. Make use of your own words in constructing your answers. 1. Probability 2. Union of two Event 3. Probability of a Union of two Events 10 Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) What I Have Learned Provide the illustration needed in each items to answer the all sub-questions which follow. 1. Draw a Venn diagram. Let Z be an event of natural numbers, let A be an event of natural numbers greater than or equal to 5 but less than 25 and let B be an event of natural numbers greater than or equal to 15 but less than 35. 1.1. What does Z in the Venn diagram represent? _________________________________________________________________ 1.2. Find . _________________________________________________________________ 1.3. Find . _________________________________________________________________ 1.4. Find . _________________________________________________________________ 2. Draw a Venn diagram. Let X be the event of red cards in a standard deck of cards and let Y be the event of face cards. 2.1. What is the universal set? _________________________________________________________________ 2.2. Find . _________________________________________________________________ 2.3. Find . _________________________________________________________________ 2.4. Find . _________________________________________________________________ 11 Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) What I Can Do At this point, you will be challenged again to formulate your own real life problem in which the application of the topic on the probability of a union of two events can be seen. After formulating your problem, identify the universal set, the events and the union of events. Assessment Given the following scenario, provide the Venn diagram to illustrate the probability of a union of two events. 1. U = {x: x is a natural number less than 20} A = {y: y is an odd number less than 20} B = {z: z is a prime number} 2. U = {x: x is a letter in an English alphabet} A = {y: y is a consonant letter} B = {z: z is a letter in the word “FOREVER”} 12 Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) 3. A = {x: x is even number less than 11} B = {y: y is an odd number less than or equal to 11} 4. U = The Milky Way. A = The Solar System B = The Stars in the Milky Way Additional Activity In this section, you are challenge to formulate your own story based on the figure provided below. The application on the probability of a union of two events has to be applied. 13 Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) Summary Probability It is defined as a ratio of how many times an event may occur as compared to the total possible outcomes. Probability of an event P(M) is equal to the total number of desired event (M) divided by the total number of possible outcomes (L). Note: o o The sum of the probabilities of all outcomes is equal to 1. Union of Two Events Given two events M and N, the union of these two events is the elements which are found in M, in N or in Both M and N In symbol, . Note: o The word or here means one or the other, or both. Probability of a Union of Two Events Given two events M and N, the probability of a union of these two ) is equal to the sum of the probability events (which can be written us of event M and the probability of event N minus the probability of (the elements occurring to both events). In this case it is assume that M and N are . not mutually exclusive events, in other words In symbol, . Note: If these two events (M and N) are mutually exclusive, then . 14 Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) Assessment: (Post-Test) Direction: Read each item carefully and choose the letter of the correct answer. 1. When choosing an “Red Ace” from a standard deck of cards, the probability is _____. B. C. D. A. 2. The sum of the probability of getting two heads and the probability of getting two tails when tossing a coin twice is equal to ____. A. B. C. D. 3. A ratio of how many times an event may occur as compared to the total possible outcomes. A. Chances B. Statistics C. Union of events D. Probability 4. In an urn, there are 15 balls in which 4 of these are in color blue, 3 are on color orange, 3 are in color violet and 5 are in color red. What is the probability of getting blue ball when ask to draw one ball from the urn? A. B. C. D. 5. (from # 5) If A is an event of red balls and B is an event of blue balls, find A. B. C. D. 6. The school principal decided to repaint the school administration building. The school custodian present four available paint colors to choose from (green, neon pink, yellow, neon red and royal blue). What is the probability that the school principal will choose green? B. C. D. A. 7. What is the probability of picking three spade face cards or two red cards? A. B. C. D. 8. Given two events Y and Z. If the and the , then this means that _____. A. the events are not mutually exclusives. B. . C. both events satisfied the condition that D. the events are intersecting at some point. For numbers 9 to 10, please refer your answers on the figures provided below. Figure 1. 15 Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) . 9. What is the probability of the complement of event A to happen? B. C. D. A. 10. What is the probability of A. B. ? C. D. 11. In tossing a coin thrice, what is the probability of getting one or two heads? C. D. A. 1 B. 12. Given to events A and B. If the and that ______. A. the events are mutually exclusives. B. both events have probability of less than 1. C. the events are intersecting. D. both events satisfied the condition which states that , then this implies . 13. In a Teachers’ Division Seminar for the new normal set-up in school opening via zoom, 90 school heads joined. 53 of the school heads voted for modular class as an alternative way of delivering lessons to the learners (event A) and 70 voted for online class (event B). Find the probability that school heads voted modular class only? A. B. C. D. Figure 2. 14. A spinner in figure 2 is equally divided into 8 parts and numbered from 1 to 8. and . What is the probability of hitting prime Let number? A. 0 B. C. D. 15. In a wedding, there are a total of 500 attendees. 380 of the attendees are friends of the bride. 330 of the attendees are friends of the groom. What is the probability that an attendee is both bride and groom’s friends? B. C. D. A. 16 Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) Key to Answers 17 Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) References Calanta, Melvin, Canonigo, Allan, Chua, Arnaldo. Cruz, Jerry, Esparrago, Mirla, Garcia, Elino, Magnaye, Aries, et. al., mathematics Learner’s Module. First edution.DepEd-IMCS: Meralco Avenvue, Pasig City, 2015. https://saylordotorg.github.io/text_introductory-statistics/s07-01-samplespaces-events-and-their.html LumenLeraning, “Computing the Probability of the Union of Two Events”. https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/cpmputingthe-probability-of-the-union-of-two-evnts/ Math is Fun, “Probability: Types of Events” accessed May 7, 2020. https://www.mathisfun.com/data/probability-events-types.html Math is Fun. “Probability” accessed https://www.mathisfun.com/data/probability.html May 7, MattePlanet, “Probability of Events” accessed May 7, https://www.mathplanet.com/education/pre-algebra/probability-andstatistic/probability-of-event 2020. 2020. “Union of Events” accessed May 7, 2020. https://probabilityformula.org/unionof-events-examples.html Statistics Libretexts, “Sample Spaces, Events, and Their Probabilities” accessed May 7, 2020. https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Book%3A_Intro ductory_Statistics_(Shafer_and_Zhang)/03%3A_Basic_Concepts_of_Probabil ity/3.01%3A_Sample_Spaces%2C_and_Their_Probabilities 18 Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com) For inquiries and feedback, please write or call: Department of Education –Learning Resources Management and Development Center(LRMDC) DepEd Division of Bukidnon Sumpong, Malaybalay City, Bukidnon Telefax: ((08822)855-0048 E-mail Address: bukidnon@deped.gov.ph Document shared on www.docsity.com Downloaded by: althea_carmella (seiccabendan@gmail.com)