T1104 [OBJECTIVE] The student will calculate theoretical and experimental probabilities, simple and compound probabilities, and independent and dependent probabilities. [MATERIALS] Student pages S416 S424 Transparencies T1114, T1116, T1118, T1120, T1122, T1124 Number cubes Red and yellow counters for students Calculators (optional) [ESSENTIAL QUESTIONS] 1. How do you find the theoretical probability of an event? 2. How do you find the experimental probability of an event? 3. What is the difference between dependent and independent probability? [GROUPING] Whole Group, Cooperative Pairs, Individual [LEVELS OF TEACHER SUPPORT] Modeling (M), Guided Practice (GP), Independent Practice (IP) [MULTIPLE REPRESENTATIONS] SOLVE, Graph, Table, Algebraic Formula, Verbal Description [WARM-UP] (5 minutes IP) S416 (Answers on T1113.) Have students turn to S416 in their books to begin the Warm-Up. Students will practice multiplying fractions to prepare for finding dependent and independent probabilities. Monitor students to see if any of them need help during the Warm- Up. Give students 3 minutes to complete the problems and then spend 2 minutes reviewing the answers as a class. {Algebraic Formula} [HOMEWORK]: (5 minutes) Take time to go over the homework from the previous night. [LESSON]: (45 55 minutes M, GP, IP)
T1105 SOLVE Problem (3 minutes GP) T1114, S417 (Answers on T1115.) Have students turn to S417 in their books, and place T1114 on the overhead. The first problem is a SOLVE problem. You are only going to complete the S step with students at this point. Tell students that during the lesson they will learn how to determine probabilities. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE} Probability (10 minutes GP, M, IP) T1114, S417 (Answers on T1115.) 3 minutes GP: As a class, read the sentences about probabilities on S417 (T1114), review the probability diagram, and fill in the blanks for the sentences about probability. {Verbal Description, Pictorial Representation} 2 minutes M: Use the following activity to review with students how to convert probabilities written as fractions into decimals and percents. {Algebraic Formula} Fractions, Decimals, and Percents (Problem 1) Step 1: Direct students attention to the fraction 1 in Problem 1. Remind students 3 that the fraction bar means divide, so students need to divide 1 by 3. Have students write this as a long division problem, 1.000 3: 3)1.000 Step 2: Ask students how many times 3 goes into 10. (3 times) Have students write a 3 above the first zero in the dividend. Ask, What is 3 times 3? (9) Have students write 9 under the 10 in the dividend. Ask, What is 10 minus 9? (1) Have students write 1 under the 9. Step 3: Demonstrate for students that, since 1 is less than 3, they need to bring down a 0 and place it next to the 1, to get 10. Repeat the process in Step 2, and then have students bring down another 0. Repeat Step 2 again. 0.333 3 )1.000 9 10 9 10 9 1
T1106 Step 4: Explain to students that since they will always have the same remainder, they can stop and write the decimal with a bar over the 3: 1 3 = 0.3 Step 5: Show students how to write the decimal as a percent by moving the decimal point two places to the right. 0.333 = 33.3% 5 minutes IP: Have students write the other three fractions on S417 (T1114) as decimals and percents. Circulate through the room to answer any questions, and then quickly check student answers. {Algebraic Formula} Theoretical Probability (8 minutes M, IP) T1116, S418 (Answers on T1117.) 3 minutes M: Have students turn to S418 in their books, and place T1116 on the overhead. As a class, read the situation and then discuss the sentences describing theoretical probability. Use the following activity to help students answer the questions. {Verbal Description, Pictorial Representation, Algebraic Formula} Theoretical Probability Step 1: Question: If the spinner landing on the number 4 is what we want to happen, what is our number of favorable outcomes? Have students count the number of 4s on the spinner. Because the number 4 appears only 1 time, there is 1 favorable outcome. Step 2: Question: If Jordan spins the spinner once, what are the possible outcomes? Have students identify the numbers on the spinner. Jordan can spin a 1, a 2, a 3, or a 4. Step 3: Question: How many total possible outcomes, or total number of things that could happen, does the spinner have? Have students count the total number of sections on the spinner. This is the number of possible outcomes for one spin.
T1107 Step 4: Question: What is the theoretical probability that Jordan will spin a 4? Have students look back at the definition of theoretical probability. Guide students to see that the number of favorable outcomes (1) is the numerator, and the total number of possible outcomes (4) is the denominator, so the theoretical probability of spinning a 4 is 1 4. 5 minutes IP: Assign students the four problems at the bottom of S418 (T1116). Give students four minutes to complete the problems, while you walk around the room and monitor. Take one minute to go over the answers. {Algebraic Formula} Experimental Probability (8 minutes M, IP) T1118, S419 (Answers on T1119.) 3 minutes M: Have students turn to S419 in their books, and place T1118 on the overhead. As a class, read the situation and then discuss the sentences describing experimental probability. Use the following activity to help students answer the questions. {Table, Verbal Description, Pictorial Representation, Algebraic Formula} Experimental Probability Step 1: Question: Look at the data Jordan collected. How many times did the spinner land on 4? Have students look at the appropriate row in the table to find the correct frequency. Jordan landed on 4 eight times. Step 2: Question: What was the total number of trials, or the total number of times the experiment was tried? Explain to students that, for the total number of trials, students need to find how many times the spinner was spun. Have students add up all the frequencies: 4 + 3 + 5 + 8 = 20. So, there were 20 total trials. Step 3: Question: What is the experimental probability of spinning a 4? Have students look back at the definition of experimental probability. Guide students to see that the number of times the event occurred (8) is the numerator, and the total number of trials (20) is the denominator, so the experimental probability of spinning a 4 is 8 20 = 2 5.
T1108 Step 4: Question: How does the theoretical probability of spinning a 4 compare to the experimental probability? Remind students that the theoretical probability of spinning a 4 was 1 4. So, the theoretical probability is less than the experimental probability. 5 minutes IP: Assign students the four problems at the bottom of S419 (T1118). Give students four minutes to complete the problems, while you walk around the room and monitor. Then take one minute to go over the answers. Be sure that students realize that sometimes the experimental and theoretical probabilities of an event will be the same, but most of the time they will not. {Verbal Description, Algebraic Formula} Independent Events (10 minutes M, IP, GP) T1120, S420 (Answers on T1121.) 3 minutes M: Have students turn to S420 in their books, and place T1120 on the overhead. As a class read the sentences about simple probability, compound probability, and probability of independent events. Use the following activity to help students answer the first four questions. You may want to allow students to use calculators to multiply the fractions. {Verbal Description, Algebraic Formula} Probability of Independent Events Step 1: Question: List all of the outcomes if you roll a number cube and flip a two-color counter. Explain to students that when they roll a number cube, they can get a 1, 2, 3, 4, 5, or 6. When they flip a two-color counter, they can get red or yellow. To find all the outcomes, have students think about what they can get on the two-color counter after they roll a 1. (If they roll a 1, they can get a yellow or a red.) Have students write down 1R and 1Y to represent these outcomes. Explain to students that they can also get red or yellow if they roll a 2, a 3, a 4, and so on. Have students write down all the outcomes: 1R, 1Y, 2R, 2Y, 3R, 3Y, 4R, 4Y, 5R, 5Y, 6R, 6Y.
T1109 Step 2: Question: What is the theoretical probability of rolling a 1 and flipping a yellow? Explain to students that they can find this out in two ways. They can count the number of outcomes that they listed which include both a 1 and a yellow (1) and put that over the total number of outcomes (12) for a probability of 1. Or, they can use the formula P(A and B) = P(A) P(B), which means 12 that the probability of two independent events happening is the product of the probabilities. The probability of the first event (rolling a 1) is 1 6. The probability of the second event (flipping a yellow) is 1. The product of the 2 two probabilities is 1 6 1 2 = 1. Either way, students get the same correct 12 answer of 1 12. Step 3: Question: What is the theoretical probability of rolling a number greater than 3, and flipping a red? Again, students can count the number of outcomes that they listed which include both a number greater than 3 and a red (3) and put that over the total number of outcomes (12) for a probability of 3 12, or 1 4. Or, students can use the formula. The probability of getting a number greater than 3 is 3 6, or 1 1 2, and the probability of getting a yellow is 2. The product of the probabilities is 1 2 1 2 = 1 4. Step 4: Question: What is the theoretical probability of not rolling a 5 and not flipping a yellow? Students can count the number of outcomes that they listed which include both a number that is not 5 and a red (5) and put that over the total number of outcomes (12) for a probability of 5 12. Or, students can use the formula. The probability of the roll not being 5 is 5 6, and the probability of the twocolor counter not being yellow is 1 2. The product of the probabilities is 5 6 1 2 = 5 12.
T1110 5 minutes IP: Divide the class into pairs and pass out one two-color counter and one number cube to each pair. Have one student flip the counter and one student roll the number cube. Ask students to do this 25 times and record their results in the table on S420. Have students answer the three questions at the bottom of the page about their data. Circulate through the room to answer questions and keep students on track. {Table, Algebraic Formula} 2 minutes GP: Bring the class back together and go over students results. {Verbal Description, Algebraic Formula} If time permits... (10 minutes GP, IP) T1122, S421 (Answers on T1123.) 6 minutes M, IP: Have students turn to S421 in their books, and place T1122 on the overhead. As a class, read the situation and then discuss the sentences describing the probability of dependent events. Use the following activity to help students answer the questions. You may want to allow students to use calculators to multiply the fractions. {Verbal Description, Algebraic Formula} Probability of Dependent Events Step 1: Question: What is the theoretical probability that Debra will pick out a yellow marble, not replace it, and pick out a green marble? Explain that students first need to find P(A), the probability of the first event. The probability of picking a yellow is 4 20 = 1 5, because there are 4 yellows out of 20 marbles. Then students need to find P(B after A), the probability of the second event happening, assuming that the first event happened. Since Debra is not replacing the first marble, there will be only 19 marbles after her first pick, but, since we assume that she picked a yellow first, there will still be 7 green marbles. So, the probability of getting a green marble second is 7 19. The probability of Debra picking a yellow marble first and a green marble second is the product 1 5 7 19 = 7 95.
T1111 Step 2: Question: What is the theoretical probability of Debra picking a blue marble from the bag, not replacing it, and then picking another blue marble from the bag? Have students find the probability of choosing the first blue marble. There are 5 blue marbles out of 20 marbles: 5 20 = 1 4. Then have students find the probability of choosing the second blue marble. Assuming that one blue marble has already been chosen, there are only 4 blue marbles out of a total of 19 marbles: 4 19. Finally, have students multiply the probabilities: 1 4 4 19 = 4 76 = 1 19. Step 3: What is the theoretical probability of Debra picking a red marble, not replacing it, and then picking a blue marble? Since there are 4 reds out of 20 marbles, the probability of getting a red is 4 20 = 1 5. There are still 5 blues, since the first marble chosen was red. There are only 19 marbles though, since one was already chosen and not put back. So, the probability of the second marble being blue is 5 19. Multiply the two probabilities together: 1 5 5 19 = 5 95 = 1 19. Step 4: Question: What is the theoretical probability of Debra picking a red marble, not replacing it, picking a blue marble, not replacing it, and then picking a yellow marble? The probability of the first marble being red is 4 20 = 1 5, because there are 4 red marbles out of 20. Once a red marble is chosen, there will be 5 blue marbles, but only 19 marbles total. The probability of choosing a blue second is 5 19. Then there will only be 18 total marbles, with 4 yellows. So the probability of the third marble being yellow is 4 18 = 2 9. Multiply the three probabilities together: 1 5 5 19 2 9 = 10 855 = 2 171. 4 minutes IP: Have students work with a partner to complete Problems 1 3 on the bottom of S421 (T1122). {Table, Algebraic Formula} SOLVE Problem (5 minutes GP) T1124, S422 (Answers on T1125.) Have students turn to S422 in their books, and place T1124 on the overhead. Remind students that the SOLVE problem is the same one from the beginning of the lesson. Complete the SOLVE problem with your students. Ask them for possible connections from the SOLVE problem to the lesson. (Step L involves multiplying fractions to find the probability of independent events.) {SOLVE, Algebraic Formula, Verbal Description}
T1112 Probability Foldable (5 minutes M, GP, IP) Pass out one sheet of colored paper to each student. Use the following activity to model for students how to fold and cut the piece of paper. Together with the students complete the foldable. {Algebraic Formula, Verbal Description} Probability Foldable Step 1: Fold the piece of paper vertically, hotdog fold. Step 2: Leave the paper folded and fold the piece of paper in half, hamburger fold, and then fold the piece of paper in half again, hamburger fold. Step 3: Open the paper up twice until you have four rectangles. Step 4: Lift the top flap up and cut on the three creases creating four flaps. Step 5: Label the outside of the foldable and complete the inside of the foldable defining the types of probability and giving examples. Create a transparency to model for students what should be written on each flap. [CLOSURE]: (5 minutes) To wrap up the lesson, go back to the essential questions and discuss them with students. How do you find the theoretical probability of an event? (Find the number of favorable outcomes, this is the numerator. Find the number of total outcomes, this is the denominator.) How do you find the experimental probability of an event? (Find the number of times the event has happened, this is the numerator. Find the number of times the trial was conducted. This is the denominator. What is the difference between dependent and independent probability? (Dependent probability means that the chances of the second event happening are dependent upon the first event. Independent means that the first event does not affect what happens on the second event.) [HOMEWORK]: Assign S423 and S424 for homework. (Answers on T1126 and T1127.)