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LESSON PLAn for Independent & Dependent Events
Method · February 2019
DOI: 10.13140/RG.2.2.22658.53447
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LESSON PLAn for Independent & Dependent Events Princess Alodia S. Alaban Craig N. Refugio, PhD I.
OBJECTIVES At the end of the period, each student should be able to: a. distinguish between independent and dependent events b. solve for the probability of independent and dependent events c. show perseverance and determination in solving the probability of independent and dependent events. II.
SUBJECT MATTER Topic: Independent and Dependent Events Materials: Cartolina, Blackboard, chalk, projector, laptop References: K-­‐12 Mathematics Learner’s Module pp. 341-­‐344 III.
PROCEDURE TEACHER’S ACTIVITY STUDENT’S ACTIVITY A. Preparation “Good Morning, class!” “Everyone, please all stand for the prayer.” “In the name of the Father, the Son, and the Holy Spirit Amen. Our father who art in heaven. Hallowed be thy name, thy kingdom come, thy will be done on earth as it is in heaven. Give us today our daily bread and forgive us our sins as we forgive those who sin against us. Do not bring us to the test and deliver us from evil. Amen.” “Please be seated. I will now check your attendance. (Checking the attendance) It seems that nobody is absent for today, Let’s have a short review. Last meeting we have discussed mutually exclusive events and not mutually exclusive events. Who can tell me what is a mutually exclusive events? All: Good Morning, ma’am A.” All: (Students are standing up.) All: “In the name of the Father, the Son, and the Holy Spirit Amen. Our father who art in heaven. Hallowed be thy name, thy kingdom come, thy will be done on earth as it is in heaven. Give us today our daily bread and forgive us our sins as we forgive those who sin against us. Do not bring us to the test and deliver us from evil. Amen.” (Students raise their hand) events? Yes, Hannah. (Students raise their hand) Hannah: Not mutually exclusive events are events that have an outcome in common. That’s right. How can we solve for the probability of the mutually exclusive events? (Students raise their hand) Yes, Dawood. Dawood: If two events, A and B, are mutually exclusive, then the probability that either A or B occurs is the sum of their probabilities. Very good. How about not mutually exclusive events? (Students raise their hand) Yes, Jay. Jay: If two events, A and B, are not mutually exclusive, then the probability that either A or B occurs is the sum of their probabilities. Correct. B. Presentation Our lesson for today is about the Independent and Dependent Events. At the end of the period, you should be able to distinguish events that are independent and dependent.you You will also learn how to solve for the probability of independent events. Who have an idea on what is the difference between independent and dependent events? Glyzha: In the word itself, independent means something that is not connected to another. So independent events are two events that is not connected to each other. Thank you Glyzha. How about dependent events? Chelsey: It is the opposite of the independent events ma’am. If independent events are two events that is not connected to each other, then dependent event is connected to another events Very good, Chelsey. Boys please read about Independent events. All Boys: Independent Events are events in which the probability of any one event occurring is unaffected by the occurrence or non-­‐occurrence of any of the other events. Thank you boys. For example, flipping a coin and rolling a die together. When we flip the coin, it does not affect the outcome of a die having a 6 or an odd number. This is also the same as passing a test and events that have no influence on each other? Yes Rica. Okay, why? That’s right. If two events, A and B, are independent, then the probability of both events occurring is the product of the probability of A and the probability of B. In symbols, P(A and B) = P(A) • P(B) Let’s try the problem that we have. Girls, please read the problem. Ready, go. Thank you girls. We have here the illustration of the box that contains the colored-­‐balls in our problem. First we are ask to find the probability that the 2 balls that are drawn are both blue. First let us find the probability of the first blue ball that is drawn from the box. If drawing first blue balls in the box is our first event, then what are we going to do? Yes Hannah. That’s right, Hannah. We simply find the probability of the (Students raise their hands) Rica: Combing your hair and finding a cat. Rica: Because It doesn’t mean that you will comb your hair, you will find your cat All Girls: Consider a box that contains 14 red balls, 12 blue balls, and 9 yellow balls. A ball is drawn at random and the color is noted and then put back inside the box. Then, another ball is drawn at random. Find the probability that: a. Both are blue. b. The first is red and the second is yellow. (Hannah raising her hand) Hannah: We find the probability of the event of the ball being randomly drawn in the box. Wherein the probability of an event is equal to number of outcomes in the event over the number of outcomes in the sample space. What is the probability that we will randomly get a blue ball in the box? Let’s have Laurish. Why did you say it is 1/12 Laurish? How about the other colors inside the box Laurish? I know. But Laurish, do you remember our lesson about Probability of simple events? That’s right. Now, the box in the problem contains not only blue balls but also red and yellow balls. Can you not consider the number of red and yellow balls in our sample space of the probability of getting blue balls? Very good. Now I will ask you again, what is the probability that we will randomly get a blue ball in the box? That’s correct. So, the probability of getting a blue ball in the box is 12/35. Let’s give Laurish 2 Claps. After the blue ball was drawn, it was put back inside the box. What is the probability that a blue ball will be drawn on the second draw? That right. It is just the same because the first blue ball that was drawn, was put back in the box. So if the person will draw again for a blue ball, then the probability will be the same. We already have the probability of the first event and the second event. Let us now use the formula to get the probability of drawing (Students raising their hands) Laurish: Ma’am I think it is 1/12. Laurish: If the person will draw a blue ball and that is only one ma’am because he/she will only draw once and it is 12 because there are 12 blue balls in the box. Laurish: They are not involved ma’am because only blue balls are asked. Laurish: Yes ma’am. The probability of simple event is equal to the number of outcomes over number of outcomes in sample space. Laurish: No ma’am because it is inside the box. So, the red balls and yellow balls are involved in the sample space in our event. Laurish: The probability of getting a blue ball in the box is 12/35. (Students Clapping) Engie: Same as the first probability of drawing of a blue ball ma’am. 12/35. = 144/1225 The probability of drawing blue balls in the box is 144/1225. Now let’s answer the second that is asked in the problem. We have here, “Find the probability of drawing first a red and the second a yellow.” We will first get the probability of drawing a red balls and then the probability of drawing a yellow ball. What is the probability that we will randomly get a red ball in the box? Why? Very well said, Zanderlie. The probability of getting first a red ball is 14/35. Now, let’s get the probability of getting a yellow ball. The probability of getting a yellow ball is 9/35, since there are 9 yellow balls out of 35 balls inside the box. Then we use the formula to get the probability of drawing red balls and yellow balls. So we have, 14 9
𝑃 𝑟𝑒𝑑 𝑎𝑛𝑑 𝑦𝑒𝑙𝑙𝑜𝑤 =
• 35 35
= 126/1225 = 18/175 The probability of drawing first a red ball and second is a yellow ball in the box is 18/175. Do you have any questions? Let’s have another example. Altogether, please read. Zanderlie: the probability that we will randomly get a red ball in the box is 14/35. Zanderlie: It is because in getting the probability of an event is equals to the number of outcomes in the over number of outcomes in the sample space. Since there are 14 red balls in the box and there are a total of 35 balls inside the box, then the answer is 14/35. All: None so far. All: Nick has 4 black pens, 3 blue pens, and 2 red pens in his school bag. Nick randomly picks two pens out of his school bag . What is the probability Yes, Marco. Very good. Then what is the probability in choosing the blue pen? Okay. That’s right. How about on the second blue pen? Then what are we going to do next? Yes, Jeadan. That’s right. So we have, 3 3
𝑃 𝐵𝑙𝑢𝑒 𝑎𝑛𝑑 𝑏𝑙𝑢𝑒 = • 9 9
9
1
=
𝑜𝑟 81
9
Do you have any questions regarding independent events? Let’s proceed to Dependent Events Everyone let’s all read together ... When we say dependent events, the outcomes of one event influence the outcome of the other event. For example, studying hard and passing the test. If you study the lessons discussed by the teacher, the possibility of passing the test is high. But, if you don’t study the lessons discussed by the teacher, then there is a high possibility of failing the test. Also, controlling your diet and having a nice curve body. If you control the amount of food you intake, then you will have a nice curve body. But if you will not control the amount of food you intake, then you will become fat . Can anyone give me an example of a dependent event? Yes, Elgie. Marco: We need to find first the probability on choosing a blue pen. (Students answering in chorus): 3/9 (Students answering in chorus): 3/9 (Students raise their hands) Jeadan: We will use the formula on finding the probability of independent events. All: None so far. All: Dependent Events -­‐ two events are dependent if the occurrence of one event does affect the occurrence of the other (e.g., random selection without replacement). Students: That’s right ma’am. (Students raise their hands) That’s right. If a person will not rob another person then he/she will not go to jail. But if he robs another person then he will go to jail. If two events, A and B, are dependent, then the probability of both events occurring is the product of the probability of A and the probability of B after A occurs. In symbols, Let’s have another problem. A box contains 7 white marbles and 7 red marbles. What is the probability of drawing a 2 white marbles and a red marble without replacement? Let us first find the probability of getting 1 white marble. The probability of an event is equal to the number of outcomes in an event over the number of outcomes in the sample space. So, the probability of getting a white marble is 7/14. Now, one marble is taken out from the box. To get the next probability of the second white marble, we only need to find the probability of getting white marble that are inside the box. The probability of the second white marble is 6/13. Why do think our answer is 6/13? (Students raising their hands) Yes, Valen. Valen: Since we already took out one white marble then the number of white marbles that are inside the box are 6 and also, our sample space is decrease to 13. Very good. We already took one marble, so the number of marbles that are in the box decreases to 13 and since the marble that we took is a white marble, then the number of marbles, we need to find the probability of getting a red marbles in the box. We already took 2 marbles in the box, then the number of our sample space will decrease to 12. Since there are 7 red marbles and 12 marbles in total, then the probability of getting a red marbles is 7/12. Then we use the formula of getting the probability of the dependent events. Since our problem is a dependent events. So we have, 7 6 7
𝑃(2 𝑤ℎ𝑖𝑡𝑒 𝑎𝑛𝑑 𝑎 𝑟𝑒𝑑) =
•
• 14 13 12
294
7
=
= 2184
52
So the probability of drawing 2 white marbles and 1 red marbles is 7/52. Let’s have another problem. Everyone, please read. Go. Thank you. So, what are we going to do first? Yes, Rachel. Very good. And what is the probability of Dominic randomly eats a banana? Yes, Bonet. That’s right. How about the apple? Yes, Glyzha. Why? Amazing. Then what are we going to do next? Yes, Romar. All: A basket contains 6 apples, 5 bananas, 4 oranges, and 5 guavas. Dominic randomly chooses one piece of fruit, eats it, and chooses another piece of fruit. What is the probability that he chose a banana and then an apple? (Students raise their hands) Rachel: First we will find the probability of the banana. (Students raise their hands) Bonet: The probability of Dominic randomly eats a banana is 5/20 or ¼. (Students raise their hands) Glyzha: Ma’am its 6/19. Glyzha: Since there are 6 apples then the number of outcomes is equal to 6 and our sample space is 19 because we can assume, that one fruit was already eaten by Dominic. (Students raise their hands) 5 6
∙ 20 19
30
3
= = 380
38
So the probability of Dominic randomly chose a banana and then an apple is 3/38. Do you understand our lesson class? Do you have any questions regarding our lesson class? 𝑃 𝑏𝑎𝑛𝑎𝑛𝑎 𝑎𝑛𝑑 𝑎𝑝𝑝𝑙𝑒 = C. Practice Let’s have a group activity. But first you go to your designated group. Group 1 will stay in here. Group two, you stay in here. Group 3 will stay in here and group 4, you stay in here. Each of your group will be given a problem that you need solve. You need to cooperate in answering the problem because I will ask someone from your group to explain. Understood? 1. A bag contains 6 black marbles, 9 blue marbles, 4 yellow marbles and 2 green marbles. A marble is randomly selected, replaced, and a second marble is randomly selected. Find the probability of selecting a black marble, then a yellow marble. 2. A box of chocolate contains 10 milk chocolates, 8 dark chocolates, and 6 white chocolates. Hanissa randomly chooses a chocolate, eats it, and then randomly chooses another chocolate. What is the probability that Hanissa chose a milk chocolate, and then, a white chocolate? 3. What is the probability of having an even number in rolling a dice twice? 4. Tom wants to pick a face card in a 52 card set. What is the probability that Tom will pick a face card thrice a row without replacement? I will only give you 5 minutes to answer. Time starts now. D. Generalization When we say independent events, what are Students: Yes ma’am Students: None so far Students: Yes ma’am. (Students participate in the activity) That’s right Jasmine. Thank you. Can somebody tell me how to solve for the probability of an independent event? How about, solving for the probability of a dependent event? That’s correct Prileen. So do you any questions and clarification regarding our lesson for today? are two events that affects the other events. Justine May: The probability of an independent events is equal to the product the probability A and probability B. Prileen: In solving for the probability of a dependent event, we multiply the probability of probability A and the probability of B following A. Students: None so far. IV.
V.
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ASSESSMENT i. Distinguish the following events if it is a dependent event or an independent event. Write DE if the event is a dependent, and IE if the events are independent. Write your answer on the space provided. _________1. Turning off the lights and winning the lottery. _________2. Murdering a person and going to jail. _________3. Winning a contest and a dog barking. _________4. Singing a high tone note and having a typhoon. _________5. Drinking an alcohol and getting drunk. ii. Answer the following problems. 1) A bag of jelly beans contains 10 red, 6 green, 7 yellow, and 5 orange jelly beans. What is the probability of randomly choosing a red jelly bean, replacing it, randomly choosing another red jelly bean, replacing it, and then randomly choosing an orange jelly bean? 2) Rene and Cris went to a grocery store to buy drinks. They chose from 10 different brands of juice drinks, 6 different brands of carbonated drinks, and 3 different brands of mineral water. What is the probability that Rene and Cris both chose juice drinks, if Rene randomly chose first and like the first brand he picked up? ASSIGNMENT What is Conditional Probability of an event?