A. Readings/Discussions: Sample Spaces and Events You learned the basic concepts on probability in grade 8. Activities such as rolling a die, tossing a coin, or randomly choosing a ball from a box which could be repeated over and over again and which have well-defined results are called experiments. The results of an experiment are called outcomes. The set of all outcomes in an experiment is called a sample space, denoted by S. For example, when a coin is tossed, there are only two possible outcomes: head (H) or tail (T). The sample space can be summarized as S = {H, T} An event is a subset of the sample space. For example, in the experiment of tossing a con, you may define an event A to be consisting of a "head" outcome. Thus, the event A is given by A = { H ). Recall that an event can be the entire sample space S. It can also be an empty set, de noted by { }or @. Simple Events: Consider rolling a die. a. "Getting a number 5" is called a simple event. b. " Getting a 6 " is also a simple event. Probability of Simple Events : If each of the outcomes in a sample space is equally likely to occur, then the probability of an event E, denoted as P(E) is given by P ( E ) = number of ways the event can occur number of possible outcomes or P( E ) = - number of outcomes in the event number of outcomes in the sample space For example, if you roll a die, the number that would come up could be 1, 2, 3, 4, 5, or 6. When the die is rolled, it is equally likely to land on one face as on any other. Therefore, the probability of getting a "5" is one out of 6. In symbol , we use P (getting a 5) = = . Always ren ember that - is the probability that any of the faces shows up. Compound Events : Events which consist of more than one outcome are called compound events. A compound event consists of two or more simple events. Example : Finding the probability of " getting a 6 and a 1 " when two dice are rolled is an event consisting of (1,6) , (6,1) as outcomes. The first die falls in 6 different ways and the second die also falls in 6 different ways. Thus, using the fundamental counting principle, the are : number of outcomes in the sample space is 6 . 6 or 36. The outcomes in the sample space((1, 1).(1,2),(1,3),(1,4),(1,5),(1,6).(2, 1),(2,2),(2,3),(2,4),(2,5).(2,6),...(6,1),(6,2),(6,3),(6,4),(6,5), (6,6) } Take note that "getting a 6 and a 1" when two dice are rolled is an event consisting of ((1,6),(6,1)} as outcomes. This is a compound event. Task 1 : Consider the situation below. Use the tree diagram given below in answering the questions that follow. A school canteen serves lunch for students. There are two types of rice, viand and drinks that the students can choose from for a set of menu consisting of one type for each. The tree diagram below shows the possible menu combinations. Rice Viand Drink chicken adobo pineapple juice Fried Rice orange juice pinakbet pineapple juice orange juice chicken adobo pineapple juice Steamed Rice orange juice pinakbet pineapple juice orange juice 1. Give the sample space combination of rice, viand, and drink. How many possible outcomes are there? 2. List the outcomes of selecting a lunch with pineapple juice. 3. How many outcomes are there for selecting any lunch with pineapple juice? 4. How many outcomes are there for selecting a lunch with steamed rice and with pineapple juice? 5. How many outcomes are there for selecting a lunch with chicken adobo and a pineapple juice? 6. How many outcomes are there for selecting a lunch with pinakbet and an orange juice? A student taking a lunch in the canteen is selected at random. 7. What is the probability that the student chose pineapple juice as a drink? 8. What is the probability that the student chose steamed rice and pineapple juice? 9. What is the probability that the student chose chicken adobo and orange juice? 10. What is the probability that the student chose pinakbet and pineapple juice? Reflect ; a. What does the tree diagram tell you? b. How did you determine the sample space? c. Differentiate an outcome from a sample space. Give another example of an outcome. d. Aside from the tree diagram , how else can you find the total number of possible outcomes? e. Describe the outcome in this situation as compared to the events that you studied in grade &Answers : 1. Sample Space : For easier listing , we use the following : fr - fried rice; sr - steamed rice; ca - chicken adobo ; p - pinakbet ; pj - pineapple juice; and oj - orange juice { (fr,ca,pj).(fr,ca,oj).(fr.p.pj).(fr,p.oj).(sr,ca,pj),(sr,ca,oj),(sr,p.pj).(sr,p.oj)} There are a total of 8 possible outcomes. 2. { (fr,ca,pj),( fr,p,pj), (sr,ca,pj).(sr,p.pj) } 3. There are 4 outcomes in selecting any lunch with pineapple juice. 4 . There are 2 outcomes for selecting a lunch with steamed rice and pineapple juice. 5. There are 2 events for selecting a lunch with chicken adobo and pineapple juice. 6. There are 2 events for selecting a lunch with pinakbet and orange juice. 7. or _ B 8 20 9. CO I NCO I NCO IN or = 10. : or Reflect : a. The tree diagram shows the total number of outcomes. 6. The sample space is obtained by listing all the outcomes that are obtained using the tree diagram C. An outcome is an element of a sample space. One example is {sr, ca,p;). d. The total number of possible outcomes can also be found using the fundamental counting principle ( multiplication rule). e. The events we studied in Grade 8 are simple events. The events in this situation are compound events. Intersection and Union of Events Intersection of Events Consider two events A and B. The intersection of events A and B is the set containing all the elements that are both in A and 8. The intersection of A and B is denoted by An B. AnB Figure 1. The intersection of events A and B is illustrated using a Venn diagram. Example 1 : Consider the experiment of tossing a die. Let A be the event of obtaining a number less than 4. Let B be the event of obtaining an odd number. Determine An B. Soution : From the given, A = { 1, 2 ,3) and B = { 1, 3 , 5 }. Thus , An B = { 1,3}. Example 2 : Consider the experiment of tossing two coins. Let A be the event of obtaining 2 heads and B be the event of obtaining at least 1 tail. Determine An B. Soution : Recall the sample space S for tossing two coins , that is , S = {HH, HT,TH,TT). From the given experiment, A = {HH) and B = {HT, TH,TT). Note that A and B do not have any element in common. Thus , An B = { }.Union of Events Consider two events A and B. The union of events A and B is the set containing all the elements that are in A, in B, or both in A and B. The union of A and B is denoted by A U B. B AUB Figure 1.2 The union of two events A and B is illustrated using a Venn diagram. Example : Consider the experiment of tossing a die. Let A be the event of obtaining 2 and B the event of obtaining an even number. Determine A U B. Solution : From the given , A = { 2 } and B = { 2,4,6 ). Thus, A U B = { 2,4,6 ). Remember: In forming the union of events, elements common to all the events are not repeated. In the example, the element 2 is an element of both events A and B but is written only once in A U B. Complement of an Event Recall that the complement of an event A is the set of all the elements in the sample space S that are not in the event A. The complement of A is denoted by A or A' or A!. This is illustrated in figure 1.3 using the Venn diagram, which you learned in grade 7. S AC Figure 1.3 The figure shows the Venn diagram of set A and its complement Ac. Example : Suppose a die is tossed. Let A be the event of obtaining an even number. Determine the complement of A. Solution : The sample space S is given by S = { 1,2,3,4,5,6 }. Since A { 2,4,6 }, A = { 1,3,5). Remember : The complement of the sample space S is the null set of , and vice versa. Task 2 : The extracurricular activities in which the senior students at Kanangga National High School participate are shown in the Venn diagram below. resExtra - curricular activities participated by senior students 1. How many students are in the senior class? 2. How many students participate in athletics? 3. How many students participate only in drama and band? 4. How many students participate in drama and athletics? 5 . How many students participate in athletics or band? 6. If a student is randomly chosen, what is the probability that the student participates in athletics or drama? 7. If a student is randomly chosen , what is the probability that the student participates only in drama and band? 8. If a student is randomly chosen, what is the probability that the student participates only in band or athletics? Answers: 1. There are 345 students in the senior class. 38+30+10+4+51+8+137+67 = 345 2. There are 159 students who participate in athletics. 4+10+8+137 = 159 3. There are 30 students who participate only in drama and band. (Note: The word "and" refers to the intersection of events, whereas "or" refers to the union of events. Drama n Band = 30+10 ( number of students who also participate in athletics).So because of the word "only", exclude 10. There are 14 students who participate in drama and athletics. Drama n Athletics = 4+10=14 5. There are 240 students who participate in athletics or band. 30+10+51+8+137+4 = 240 . Since there are 227 students who participate in athletics or drama and there are a total of 345 students in the senior high, therefore the probability that the student participates in athletics or drama is 345 7. The probability that the student participates only in drama and band is 30 345 . 8. There are 240 students who participate in band or athletics. But there are 44 who participate drama. Therefore, the probability that the student participates only in band or athletics is 196 345 Illustrative Example : The Venn diagram below shows the probabilities of grade 10 students joining either soccer ( S ) or basketball ( B ). 0.4( 0.3 ) 062 Use the Venn diagram to find the probabilities. a. P(B) b. P (S) C. P(BnS) d. P(BUS)e. P(B'nS') Discussion of the Solution: Actually, the diagram does not show or represent the entire sample space for B and S. What is shown are the probabilities. a. To find the P(B) , we will add the probability that only B occurs to the probability that B and S occur to get 0.4+0.3 = 0.7. So, P(B) = 0.7 . b. Similarly, P(S) = 0.2 + 0.3 = 0.5. c. Now , P(BnS) is the value 0.3 in the overlapping region. d. P(BUS) = 0.4 + 0.3 + 0.2 = 0.9 e. P(B'nS') = 1 - P(BUS) = 0.1 Remember: The probability of an event A must be between 0 and 1. inclusive , that is , OS P(A) $ 1. A large value of probability means that the event is most likely to occur. . If the probability is 1, the event is certain to happen. If the probability is 0, the event will never happen. B. Exercises Exercise 1 : Answer the following. 1. Consider tossing two coins. a. Make a tree diagram of the sample space for tossing two coins. b. List the sample space. How many outcomes are there? c. What is the probability of getting exactly 2 heads? 2. If P(M) = 0.35, P(N) = 0.5, and P(MON) = 0.2, find P(MUN). 3. In a mathematics seminar, 15 students are in grade 7, 10 are in grade 8, 20 are in grade 9, and 30 are in grade 10. If a student will be chosen at random, what is the probability of selecting a grade 10 student? Exercise 2 : Answer the following. 1. The probability that a student owns a mobile cellphone is 0.83, and the probability that he owns a laptop computer is 0.72. The probability that he owns both is 0.68. a. Draw a Venn diagram showing these probabilities. b. What is the probability that he will own a mobile phone or a laptop computer? 2. Use the Venn diagram to find the following probabilities. 0.09/ 0.06 034 051 a. P(X) b. P(A) C. P(AUX) d. P(AnX) e. P(AUX') f. P(AnX)C. Assessment/Application/Outputs (Please refer to DepEd Order No. 31, s. 2020) Part 1. Multiple Choice : Answer each question. Choose the letter of the correct answer. 1.Which of the following is a subset of a sample space? A event C. sample point D. outcome B. experiment 2. What is the probability of the entire sample space? D. 1 A 0 B. 0.5 C. 0.99 3. Which of the following is the complement of getting a king from an ordinary deck of playing carls? A getting a heart B.getting a queen C. getting a non - king card D. not getting a heart 4. I a box containing 15 mint candies and 10 chocolates, what is the probability of getting a mint candy if 1 candy is chosen at random? A _ B. = C. D. 5. 1 P(A) = 0.2 , P(B) = 0.1 , and P(An B) = 0.07 , what is P(AUB)? A 0.13 B. 0.23 C. 0.3 D. 0.4 Part II. Answer what is being asked. 6. Use the following situation: The international club of school has 105 members,many of whom speak multiple languages. The most commonly spoken languages in the club are English, Korean, and Chinese. Use the Venn diagram below to determine the following: Korean chinese 35 English a. number of students who can speak English and Korean. b. number of students who can either speak Korean or Chinese. C. the probability of selecting a student who does NOT speak English. d. the probability of selecting Korean and English but NOT Chinese. e. the probability of selecting a student who neither speaks the three languages. D. Suggested Enrichment/Reinforcement Activitylies : Directions : Do as indicated. 1. Describe the union and intersection of two events. Give examples