Math Grade 8 Part 2

A.M. = 28 ∑f = 40 ∑fd =16 i=5 Mean = A.M + ∑(fd) i ∑f Mean = 28 + 16 5 40 Mean = 28 + 16(5) 40 Mean = 28 + 80 40 Mean = 28 + 2 Mean = 30 Therefore, the mean of the mid-year test scores is 30. What have you observed? This implies that even if you use class marks or coded deviation, the results are the same.2. Median for Grouped Data The median is the middle value in a set of quantities. It separates an ordered set of data into two equal parts. Half of the quantities is located above the median and the other half is found below it, whenever the quantities are arranged according to magnitude (from highest to lowest.) In computing for the median of grouped data, the following formula is used: Median = lbmc + ∑f − <cf i 2 fmcwhere: lbmc is the lower boundary of the median class; f is the frequency of each class; <cf is the cumulative frequency of the lower class next to the median class; fmc is the frequency of the median class; and i is the class interval. The median class is the class with the smallest cumulative frequency greater than or ∑fequal to 2 . The computed median must be within the median class. 529

Illustrative Example:Direction: Calculate the median of the Mid-year Test Scores of students in Filipino. Mid-year Test Scores of Students in Filipino Score Frequency 41 – 45 1 36 – 40 8 31 – 35 8 26 – 30 14 21 – 25 7 16 – 20 2Solution: Frequency lb <cf Score 1 40.5 40 41 – 45 8 35.5 39 36 – 40 8 30.5 31 31 – 35 14 25.5 23 Median Class 26 – 30 7 20.5 9 21 – 25 2 15.5 2 16 – 20 i=5 ∑f = 40 Median = lbmc + ∑f − <cf i 2 ∑f 40 fmc 2 2a. = = 20 The 20th score is contained in the class 26-30. This means that the median falls within the class boundaries of 26-30. That is, 25.5-30.5 b. <cf = 9 c. fmc = 14 d. lbmc = 25.5 e. i = 5Solution: ∑f 2 Median = lbmc + − <cf i fmc Median = 25.5 + 40 − 9 5 2 14 530

Median = 25.5 + 20 − 9 5 14Median = 25.5 + 11 5 14Median = 25.5 + 55 14Median = 25.5 + 3.93Median = 29.43Therefore, the median of the mid-year test scores is 29.43. (Note: The median 29.43 falls within the class boundaries of 26-30 which is 25.5-30.5)3. Mode for Grouped Data The mode of grouped data can be approximated using the following formula:Mode = lbmo + D1 i D1 + D2where: lbmo is the lower boundary of the modal class; D1 is the difference between the frequencies of the modal class and the next upper class; D2 is the difference between the frequencies of the modal class and the next lower class; and i is the class interval.The modal class is the class with the highest frequency.Illustrative Example:Directions: Calculate the mode of the mid-year test scores of students in Filipino.Mid-year Test Scores of Students in Filipino Score Frequency41 – 45 136 – 40 831 – 35 826 – 30 1421 – 25 716 – 20 2 531

Solution: Score Frequency lb 41 – 45 1 40.5 36 – 40 8 35.5 31 – 35 8 30.5 26 – 30 14 25.5 Modal Class 21 – 25 7 20.5 16 – 20 2 15.5Since class 26-30 has the highest frequency, therefore the modal class is 26-30. lbmo = 25.5 D1 = 14 – 8 = 6 D2 = 14 – 7 = 7 i=5 Mode = 25.5 + D1 i D1 + D2 Mode = 25.5 + 6 5 6+ 7 Mode = 25.5 + 6 5 13 Mode = 25.5 + 30 13 Mode = 25.5 + 2.31 Mode = 27.81 Therefore, the mode of the mid-year test scores is 27.81. 532

Illustrative Example: Height of Nursing Students in Our Lady of Piat CollegeHeight (cm) Frequency <cf170-174 8 50165-169 10 42160-164 11 32155-159 11 21150-154 10 10 (Note: The given data have two classes with the highest frequency; therefore, the firstformula in solving the mode is not applicable.)Solutions: ∑(fX) = 8075 = 161.5 a. Mean = ∑f 50 Mean = 161.5 b. M∑ef dia5n0 2 = 2 = 25 The smallest cumulative frequency greater than 25 occurs in the class 160-164. This means that the median falls within the class boundaries of 160-164.That is, 159.5-164.5<cf = 21fmc = 11 ∑flbmc = 159.5 2i = 5Median = lbmc + − <cf i fmcMedian = 159.5 + 25 − 21 i 11Median = 159.5 + 4 5 11Median = 159.5 + 4(5) 11Median = 159.5 + 20 11Median = 159.5 + 1.82Median = 161.32 533

Were you able to learn different formulas in solving the mean, median, andmode of grouped data? In the next activity, try to apply those important notes ingetting the value of the mean, median, and mode of grouped data.Activity 4 LET’S SOLVE IT…Directions: Calculate the mean, median and mode of the weight of IV-2 students. Write your complete solutions and answers on a sheet of paper. Weight of IV-2 Students Weight in kg Frequency 75 – 79 1 70 – 74 4 65 – 69 10 60 – 64 14 55 – 59 21 50 – 54 15 45 – 69 14 40 – 44 1 Mean = _______________________ Median = _______________________ Mode = _______________________QU?E S T I ONS 1. How did you find the mean, median, and mode of the set of data? 2. What comparisons can you make about the three measures obtained? 3. What have you learned and realized while doing the activity? 534

Have you solved the mean, median, and mode easily with your partner? Wereyou able to apply the notes on how to calculate the mean, median, and mode? Dothe next activity by yourself.Activity 5 ONE MORE TRY…Directions: Calculate the mean, median, and mode of the given grouped data. Pledges for the Survivors of Typhoon Pablo Pledges in Pesos Frequency 9,000 – 9,999 4 8,000 – 8,999 12 7,000 – 7,999 13 6,000 – 6,999 15 5,000 – 5,999 19 4,000 – 4,999 30 3,000 – 3,999 21 2,000 – 2,999 41 1,000 – 1,999 31 0 – 999 14QU?E S T I ONS 1. What is the class interval of the given frequency distribution table? 2. How many pledges are there for the survivors of the typhoon? 3. Determine the following: a. Class mark of the pledges having the highest number of donors b. Median class c. Modal class 4. How did you determine the mean, median, and the mode of the given data set? How about the lower boundary of the median class of the pledges? 5. What is the lower boundary of the median class of the pledges in pesos? 6. What is the lower boundary of the modal class? 7. What is the modal score of the pledges in pesos? 535

REFLECTION I_n______t_____h________i_____s___________________l______e___________s____________s__________o______________n_________________,______________I____________________h_______________a_____________v____________e___.________________u______________n_______________d_______________e___________r__________s________t_______o_______________o_____________d______________________t__________h____________a____________t_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 536

WWhhaatt ttoo UUnnddeerrssttaanndd Reflect how you were able to develop a concept out of the activities you have studied. The knowledge gained here will further help you understand and answer the next activities. After doing the following activities, you should be able to answer the following question: How are the measures of central tendency for grouped data used in solving real-life problems and in making decisions?Activity 6 WE CAN DO IT…1. Below are the scores of 65 students in a Mathematics test. Score f X d fd <cf55 – 58 251 – 54 447 – 50 543 – 46 639 – 42 1035 – 38 1331 – 34 827 – 30 623 – 26 619 – 22 215 – 18 211 – 14 1a. Complete the table by filling in the values of X (the class marks or midpoints), d(deviation), fd, and <cf (cumulative frequency). Explain how you arrived at your answers.b. Find the mean, median, and the mode of the set of data.c. How would you compare the mean, the median, and the mode of the set of data?d. Which measure best represents the average of the set of data? Why?2. Which is the most appropriate measure of central tendency for the set of data? Why?3. Is it always necessary to group a set of data when finding its mean, median, or mode? Why? 537

What new insights do you have about solving measures of central tendencyof grouped data? What do you realize after learning and doing different activities? Let’s extend your understanding. This time, apply what you have learned inreal life by doing the tasks in the next section.WWhhaatt ttooTTrraannssffeerr Your goal in this section is to apply your learning to real-life situations. You will begiven a practical task which will demonstrate your understanding of solving measures ofcentral tendency of grouped data.Activity 7 LET’S APPLY IT…. Prepare some power saving measures. Gather data from your classmates or peerswhich may include the following: electric bills, electric appliances, and the estimated time ofusage. Use the data and different statistical measures obtained for analysis and coming upwith power-saving measures.Make use of this rubric in assessing your work or output. RUBRIC ON GROUP TASK 4 3 21Understanding I/we I/we I/we I/we of Task demonstrated demonstrated an in-depth substantial demonstrated demonstrated understanding understanding of the content, of the content gaps in our minimal processes, and and task, even demands of the though some understanding of understanding task. supporting ideas or details the content and of the content. may have been overlooked or task. misunderstood. 538

Completion of I/we fully I/we I/we completed I/we attempted Task achieved the accomplished most of the task. to accomplish purpose of the the task. the task, butCommunication task, including I/we with little or no of Findings thoughtful, I/we communicated success. insightful communicated our ideas andGroup Process interpretations our findings findings. I/we did not and conjectures. effectively. finish the We worked investigation I/we We worked well together some and/or were communicated together most of the time. not able to our ideas of the time. We Not everyone communicate and findings usually listened contributed our ideas very effectively, to each other equal efforts to well. raised and used each the task. interesting and other's ideas. We really provocative We might have did not pull questions, and worked more together or went beyond productively as work very what was a group. productively expected. as a group. Not everyone We used all contributed of our time to the group productively. effort. Everyone was Some people involved and did more work contributed than others. to the group OR Nobody process and worked very product. well in the group.Problem Solving Problems did We worked not deter us. We together to were proactive overcome and worked problems we together to solve encountered. problems.Adopted from Intel Teach Elements (Assessment on 21st Century Classroom) In this section, your tasks were to cite real-life situations and formulate andsolve problems involving measures of central tendency of grouped data. How did you find the performance task? How did the task help you see thereal world application of measures of central tendency of grouped data? 539

REFLECTION I_n______t_____h________i_____s___________________l______e___________s____________s__________o______________n_________________,______________I____________________h_______________a_____________v____________e___.________________u______________n_______________d_______________e___________r__________s________t_______o_______________o_____________d______________________t__________h____________a____________t_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 540

SUMMARY/SYNTHESIS/GENERALIZATION: This lesson was about measures of central tendency of grouped data. The lessonprovided you with opportunities to describe on how to solve the mean, median, and mode ofthe given grouped data. Moreover, you were given the chance to apply the important noteson how to solve the mean, median, and mode of the given grouped data and to demonstrateyour understanding of the lesson by doing a practical task. 541

4Lesson Measures of Variability of Grouped DataWWhhaatt ttoo KKnnooww Start the lesson by assessing your knowledge of the different mathematics concepts previously studied and your skills in performing mathematical operations. These knowledge and skills may help you in understanding Measures of Variability of Grouped Data. As you go through this lesson, think of the following important question: How are the measures of variability of grouped data used in solving real-life problems and in making decisions? To find out the answer, perform each activity. If you find any difficulty in answering the exercises, seek the assistance of your teacher or peers or refer to the modules you have studied earlier.Activity 1 LET’S TRY THIS!Directions: Complete the frequency distribution table by finding the unknown values. Write your complete solutions and answers on a piece of paper. Scores of Grade 8 Avocado Students in the 4th Periodic Test in Mathematics Score Frequency Class fX (X − x) (X − x)2 f(X − x)2 ( f ) Mark (X) 46 – 50 41 – 45 2 36 – 40 31 – 35 9 26 – 30 21 – 25 13 i= 11 10 5 ∑f(X − x)2 = ∑f = ∑fX = 542

QU?E S T I ONS 1. How did you determine the unknown values in the frequency distribution table? 2. What is the class size? 3. What is the ∑fX? 4. What is the value of the mean in the given distribution table? 5. What is the upper class boundary of the top interval? What about the lower class boundary of the bottom interval? 6. What is the range? 7. What is the variance of the given distribution table? 8. How would you find the variance? 9. What is the standard deviation? 10. How would you solve for the standard deviation? Were you able to complete the frequency distribution table? Were you ableto find the unknown values in the frequency distribution table? In the next activity,you will calculate the range, variance, and standard deviation of a given data set.Activity 2 GO FOR IT…Directions: The frequency distribution below shows the number of mistakes 50 students made in factoring 20 quadratic equations. Use the table to answer the questions that follow. Write your complete solutions and answers on a piece of paper. Number of Mistakes Made by 50 Students in Factoring 20 Quadratic Equations Number of Mistakes Frequency X 18 – 20 2 15 – 17 5 12 – 14 6 9 – 11 10 6–8 15 3–5 8 0–2 4 543

QU?E S T I ONS 1. What is the total frequency of the given data set? 2. Complete the frequency distribution table. What is ∑fX? 3. How would you find the mean of the given data set? 4. What is the mean of the set of data? 5. What is the upper class boundary of the top interval? 6. What is the lower class boundary of the bottom interval? 7. What is the range? 8. Find the variance and standard deviation of the set of data. 9. How are the range, variance, and standard deviation used in interpreting the set of data?WWhhaatt ttoo PPrroocceessss How did you find the previous activity? Were you able to find the unknownmeasures/values? Are you ready to perform the next activity? Will you be able to find themean, range, variance, and standard deviation of a set of data such as the grades or testscores? Before proceeding to these activities, read first some important notes on how tocalculate the range, variance, and standard deviation of grouped data. To find the range, variance, and standard deviation of grouped data, take note of thefollowing:1. The Range of Grouped Data The range is the simplest measure of variability. The range of a frequency distribution is simply the difference between the upper class boundary of the top interval and the lower class boundary of the bottom interval. Upper Class Boundary Lower Class Boundary Range = of the Highest Interval – of the Lowest IntervalIllustrative Example: Solve for the range: Scores in the Second Periodical Test of I – Faith in Mathematics I Scores Frequency 46 – 50 1 41 – 45 10 36 – 40 10 31 – 35 16 26 – 30 9 21 – 25 4 544

Solutions: Upper Class Limit of the Highest Interval = 50 Upper Class Boundary of the Highest Interval = 50 + 0.5 = 50.5 Lower Class Limit of the Lowest Interval = 21 Lower Class Boundary of the Lowest Interval = 21 − 0.5 = 20.5Range = Upper Class Boundary – Lower Class Boundary of the Highest Interval of the Lowest Interval Range = 50.5 – 20.5 Range = 30 Therefore, the range of the given data set is 30.2. The Variance of Grouped Data (σ2) Variance is the mean of the square of the deviations from the mean of a frequencydistribution. For large quantities, the variance is computed using frequency distributionwith columns for the midpoint value, the product of the frequency and midpoint valuefor each interval, the deviation and its square, and the product of the frequency and thesquared deviation.To find variance of a grouped data, use the formula: σ2 = ∑f(X − x)2 ∑f − 1 where; f = class frequency X = class mark x = class mean ∑f = total number of frequency In calculating the variance, do the following steps:1. Prepare a frequency distribution with appropriate class intervals and write the corresponding frequency ( f ).2. Get the midpoint (X) of each class interval in column 2.3. Multiply frequency ( f ) and midpoint (X) of each class interval to get fX.4. Add fX of each interval to get ∑fX.5. Compute the mean using x = ∑∑fXf .6. Calculate the deviation (X − x ) by subtracting the mean from each midpoint.7. Square the deviation of each interval to get (X − x )2.8. Multiply frequency ( f ) and (X − x )2. Find the sum of each product to get ∑f(x− x)2.9. Calculate the variance using the formula σ2 = ∑f(X − x)2 ∑f − 1 545

Illustrative Example: Find the variance of the given data set: Scores in the Second Periodical Test of I – Faith in Mathematics I Scores Frequency 46 – 50 1 41 – 45 10 36 – 40 10 31 – 35 16 26 – 30 9 21– 25 4Solutions:Scores Frequency Class fX (X − x) (X − x)2 f (X − x)2 (f) Mark (X)46 – 50 1 48 48 13.4 179.56 179.56 43 430 8.4 70.56 705.641 – 45 10 38 380 3.4 11.56 115.6 33 528 -1.6 2.56 40.9636 – 40 10 28 252 -6.6 43.56 392.04 23 92 -11.6 134.56 538.2431 – 35 16 ∑fX = 1,730 ∑f(X − x)2= 1,97226 – 30 921 – 25 4 i=5 ∑f = 50 Mean (x) = ∑fX = 1,730 = 34.60 ∑f 50 σ2 = ∑f(X − x)2 ∑f − 1 σ2 = 1,972 50 − 1 σ2 = 1,972 = 40.2448 ≈ 40.24 49 Therefore, the variance(σ2) is 40.24. 546

3. Standard Deviation (s) The standard deviation is considered the best indicator of the degree of dispersion among the measures of variability because it represents an average variability of the distribution. Given the set of data, the smaller the range, the smaller the standard deviation, the less spread is the distribution. To get the value of the standard deviation (s), get the square root of the variance (σ2): s = √σ2Illustrative Example: Refer to the previous example. Get the square root of the value of variance: s = √σ2 s = √40.24 s = 6.34 Therefore, the standard deviation of the Scores in the Second Periodical Test of I – Faith in Mathematics I is 6.34. Were you able to learn the formulas in solving the range, variance, and standard deviation of grouped data? In the next activity, try to apply the important notes in getting the value of the range, variance, and standard deviation of grouped data.Activity 3 LET’S APPLY IT…Directions: Calculate the range, variance, and standard deviation of the Weekly Allowance of Students in Binago School of Fisheries. Write your complete solutions and answers on a sheet of paper. Weekly Allowance of Students in Binago School of Fisheries Weekly Allowance Frequency (in Pesos) 500-549 2 450-499 3 Range = ____________________________ 400-449 Variance (σ2) = ________________________ 350-399 1 Standard Deviation (s) = _________________ 300-349 3 4 250-299 14 200-249 12 150-199 21 100-149 10 547

QU QU? ES TIO 1. NS NS How did you find the range, variance, and standard deviation? 2. What you can say about the value of range and variance? 3. What you can say about the standard deviation? 4. What have you learned and realized while doing the activity? Were you able to solve the range, variance, and standard deviation easily with your seatmate? Were you able to apply the notes on how to calculate the range, variance, and standard deviation? Do the next activity by yourself. Activity 4 CHALLENGE PART… Directions: Calculate the range, variance, and standard deviation of the given grouped data. Pledges for the Victims of Typhoon Pablo Pledges in Pesos Frequency 9,000 – 9,999 4 8,000 – 8,999 12 7,000 – 7,999 13 6,000 – 6,999 15 5,000 – 5,999 19 4,000 – 4,999 30 3,000 – 3,999 21 2,000 – 2,999 41 1,000 – 1,999 31 0 – 999 14 ?E S T I O 1. What is the ∑fX? 2. What is the value of the mean in the given distribution table? 3. What is the upper class boundary of the top interval? What about the lower class boundary of the bottom interval? 4. What is the range? 5. What is the variance of the given distribution table? 6. How would you find the variance? 7. What is the standard deviation? 8. How would you solve for the standard deviation? 9. What have you learned from the given activity? 548

REFLECTION W____h_______a_______t___________I_______________h______________a____________v____________e_________________l_______e____________a___________r___________n______________e____________d____________________s_________o____________________f_________a____________r________.______.______.___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________._________________________________________________ 549

WWhhaatt ttoo UUnnddeerrssttaanndd Reflect on how you were able to develop a concept out of the activities you have studied. The knowledge gained here will further help you understand and answer the next activities. After doing the following activities, you should be able to answer the following question: How are the measures of variability of grouped data used in solving real-life problems and in making decisions?Activity 5 LET’S CHECK YOUR UNDERSTANDING…1. Below are the scores of 65 students in a Mathematics testScore f X fX (X − x) (X − x)2 f (X − x)255 – 58 251 – 54 447 – 50 543 – 46 639 – 42 1035 – 38 1331 – 34 827 – 30 623 – 26 619 – 22 215 – 18 211 – 14 1a. Complete the table by filling in the values of X (the class marks or midpoints), (X − x), (X − x)2, and f(X − x)2. Explain how you arrived at your answer.b. Find the range, variance and standard deviation of the set of data.c. What you can say about the standard deviation?d. Which measure is considered appropriate? Why?2. Which among the range, variance, and standard deviation is the most appropriate measure of variability? Why?3. Is it always necessary to group a set of data when finding its range, variance, and standard deviation? Why? 550

What new insights do you have about solving measures of variability of grouped data? What do you realize after learning and doing different activities? Now, you can extend your understanding by doing the tasks in the next section.WWhhaatt ttooTTrraannssffeerr Demonstrate your understanding on measures of central tendency and measures of variability through products that reflect meaningful and relevant problems/situations. Create a scenario of the task in paragraph form incorporating GRASPS: Goal, Role,Audience, Situation, Product/Performance, Standards. G: Make a set of criteria for a scholarship grant based on monthly family income and scholastic performance. R: Barangay Social Worker A: Local NGO S: An NGO in the locality will grant scholarship to qualified and deserving scholars P: Criteria S: Justification, Accuracy of data, Clarity of Presentation 551

REFLECTION I_n______t_____h________i_____s___________________l______e___________s____________s__________o______________n_________________,______________I____________________h_______________a_____________v____________e___.________________u______________n_______________d_______________e___________r__________s________t_______o_______________o_____________d______________________t__________h____________a____________t_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 552

SUMMARY/SYNTHESIS/GENERALIZATION: This lesson was about measures of variability of grouped data. The lesson provided youwith opportunities to describe how to solve the range, variance and standard deviation of thegiven grouped data. Moreover, you were given the chance to apply the given important noteson how to solve the range, variance, and standard deviation of the given grouped data and todemonstrate your understanding of the lesson by doing a practical task.POST ASSESSMENT: Take the test you took at the beginning of this module. Answer the questions in the Pre-assessment.GLOSSARY OF TERMSAverage Deviation or Mean Deviation - The dispersion of a set of data about the mean ofthese dataClasses - categories for grouping dataClass Mark - the midpoint of a classClass Width - the difference between the lower class boundary of the given class and thelower boundary of the next higher class.Data - a collection of facts or information from which conclusions may be drawn.Frequency - the number of data values in a classFrequency Distribution - a listing of classes and their frequencyLower Class Boundary - the smallest value that can go in a classMeasure of Central Tendency - the score or value where all the other values in a distributiontend to clusterMeasure of Variability - is a measure that describes how spread out or scattered a set ofdata. It is also known as measure of dispersion or measure of spread.Mean - sum of measures x divided by the number N of measures in a variable. It is symbol-ized as x (read as x bar).Median - the middle entry or term in a set of data arranged in numerical order (either increas-ing or decreasing). If the number of term is even, the median is the mean of the two middlescores. 553

Mode - the measure or value which occurs most frequently in a set of data. It is the value withthe greatest frequencyRange - the simplest measure of variability. It is the difference between the largest value andthe smallest value.Relative Frequency - the ratio of the frequency of a class to the total number of pieces of dataRelative Frequency Distribution - a listing of classes and their relative frequenciesStandard Deviation - is the average amount by which individual data items in a set of datadiffer from the arithmetric mean of all the data in the set. The standard deviation is the square-root of the variance.Upper Class Boundary - the largest value that can go in a classVariance - is the square of the standard deviation.REFERENCES AND WEBSITE LINKS USED IN THIS MODULECanonigo, A. (2012) Statistics, K – 12 Facilitators GuideMath IV, Project EASEAmsco_Integrated Algebra, StatisticsJavier, S. CASIO FX – 991ES PLUS HandbookPogoso, C., Montana, R, Introductory Statisticshttp://www.picturesof.net/search_term_pages/meat.htmlhttp://www.stockfresh.com/image/289009/cartoon-kidshttp://www.fotosearch.com/photos-images/garment-factory.htmlhttp://www.equinoxlab.com/http://www.themall.ph/thumbs/images/2012/10/25/get1017125146/1_13511564701025145x217.JPGhttp://media3.picsearch.com/is?tqgS6MH-ZoNLoIkQZGJu2Qhkc_04U6Jd88KruCCxpz4http://cdn7.fotosearch.com/bthumb/CSP/CSP428/k4285733.jpg 554

8 Mathematics Learner’s Module 11This instructional material was collaboratively developed andreviewed by educators from public and private schools,colleges, and/or universities. We encourage teachers andother education stakeholders to email their feedback,comments, and recommendations to the Department ofEducation at [email protected] value your feedback and recommendations. Department of Education Republic of the Philippines

Mathematics – Grade 8Learner’s ModuleFirst Edition, 2013ISBN: 978-971-9990-70-3 Republic Act 8293, section 176 indicates that: No copyright shall subsist inany work of the Government of the Philippines. However, prior approval of thegovernment agency or office wherein the work is created shall be necessary forexploitation of such work for profit. Such agency or office may among other things,impose as a condition the payment of royalties. The borrowed materials (i.e., songs, stories, poems, pictures, photos, brandnames, trademarks, etc.) included in this book are owned by their respectivecopyright holders. The publisher and authors do not represent nor claim ownershipover them.Published by the Department of EducationSecretary: Br. Armin Luistro FSCUndersecretary: Dr. Yolanda S. Quijano Development Team of the Learner’s Module Consultant: Maxima J. Acelajado, Ph.D. Authors: Emmanuel P. Abuzo, Merden L. Bryant, Jem Boy B. Cabrella, Belen P. Caldez, Melvin M. Callanta, Anastacia Proserfina l. Castro, Alicia R. Halabaso, Sonia P. Javier, Roger T. Nocom, and Concepcion S. Ternida Editor: Maxima J. Acelajado, Ph.D. Reviewers: Leonides Bulalayao, Dave Anthony Galicha, Joel C. Garcia, Roselle Lazaro, Melita M. Navarro, Maria Theresa O. Redondo, Dianne R. Requiza, and Mary Jean L. Siapno Illustrator: Aleneil George T. Aranas Layout Artist: Darwin M. Concha Management and Specialists: Lolita M. Andrada, Jose D. Tuguinayo, Jr., Elizabeth G. Catao, Maribel S. Perez, and Nicanor M. San Gabriel, Jr.Department of Education-Instructional Materials Council Secretariat (DepEd-IMCS) 2nd Floor Dorm G, PSC Complex, Meralco Avenue.Office Address: Pasig City, Philippines 1600Telefax: (02) 634-1054, 634-1072E-mail Address: [email protected]

Table of Contents Unit 4Module 11: Introduction to Probability ....................................................555 Module Map....................................................................................................... 556 Pre-Assessment ................................................................................................ 557 Learning Goals .................................................................................................. 561 Lesson 1: Basic Concepts of Probability....................................................... 562 Activity 1 ........................................................................................................ 562 Activity 2 ........................................................................................................ 563 Activity 3 ........................................................................................................ 654 Activity 4 ........................................................................................................ 566 Activity 5 ........................................................................................................ 567 Activity 6 ........................................................................................................ 570 Activity 7 ........................................................................................................ 570 Activity 8 ........................................................................................................ 572 Activity 9 ........................................................................................................ 573 Activity 10 ...................................................................................................... 576 Activity 11 ...................................................................................................... 577 Activity 12 ...................................................................................................... 577 Activity 13 ...................................................................................................... 578 Activity 14 ...................................................................................................... 579 Activity 15 ...................................................................................................... 580 Activity 16 ...................................................................................................... 581 Activity 17 ...................................................................................................... 583 Summary/Synthesis/Generalization ............................................................... 585 Glossary of Terms ........................................................................................... 585 References and Website Links Used in this Module ..................................... 586 iii

INTRODUCTION TO PROBABILITYI. INTRODUCTION AND FOCUS QUESTIONS Do you think it is possible for you to determine the chance of occurrence of an event? Have you at a certain time asked yourself the following questions?What are the possible What are my chances of getting theroutes that I can take correct answer in a True/False-type in going to school? question? Multiple choice-type of question? How likely is it that I Should I bring my umbrellawill be called to recite tomorrow? in our math class Will I probably win in today? this game?How do you deal with these questions? Were you able to answer them with certainty? In this module, you will learn more about the rich applications of the fundamentalcounting principles and probability. Remember to search for the answer to the followingquestions: How is the number of occurrences of an event determined? How doesknowledge of finding the likelihood of an event help you in your daily life?II. LESSONS AND COVERAGE In this module, you will examine the aforementioned questions when you study the following lessons: Lesson 1 – Basic Concepts of Probability Lesson 2 – Probability of an Event: Experimental Probability and Theoretical Probability Lesson 3 – Organizing Outcomes of an Event and the Fundamental Counting Principles Lesson 4 – Problems Involving Probabilities of Events 555

In these lessons, you will learn to: Lesson 1 Define experiment, outcomes, sample space, and event. Lesson 2 Explain and interpret the probability of an event; andLesson 3 Differentiate between an experimental probability and a theoreticalLesson 4 probability. Count the number of occurrences of an outcome in an experiment and organize them using a table, tree diagram, systematic listing, and the fundamental counting principles. Solve simple problems involving probabilities of events. MMoodduullee MMaapp Here is a simple map of the lessons that will be covered in this module. Probability Basic ConceptsExperimental Theoretical Probability Probability Counting Techniques Problems Involving Probability of Simple Events 556

III. PRE-ASSESSMENT Find out how much you already know about this module. Write the letter that youthink best answers the question. Please answer all items. After taking this short test, youwill see your score. Take note of the items that you were not able to answer correctlyand find for the right answers as you go through this module.1. Which of the following DOES NOT belong to the group? a. Chance b. Interpretation c. Possibilities d. Uncertainty2. All the possible outcomes that can occur when a coin is tossed twice are listed in the box. What is the probability of having a head? a. 1 HH TH b. 4 TT HT c. 1 10 d. 2 PISO 3 4 APOLINARIO MABINI ANDRES BONIFACIO 2013 13. The local weather forecaster said that there is a 20% chance of rain tomorrow. What is the probability that it will not rain tomorrow? a. 0.2 b. 0.8 c. 20 d. 804. A relative-frequency distribution for scores in a 5-item test is provided in the table below. Score 0 1 2 3 4 5 Relative 0.105 0.316 0.352 0.180 0.043 0.004Frequency Suppose that the passing score is 4, what is the probability that a randomly selected student failed the quiz? a. 0.047 b. 0.575 c. 0.773 d. 0.953 557