Probability and Statistics

Transcription

1 Probability and Statistics Activity: TEKS: Complements Come Easily! (6.9) Probability and statistics. The student uses experimental and theoretical probability to make predictions. The student is expected to: (A) construct sample spaces using lists and tree diagrams; and (B) find the probabilities of a simple event and its complement and describe the relationship between the two. (6.13) Underlying processes and mathematical tools. The student uses logical reasoning to make conjectures and verify conclusions. The student is expected to: (A) make conjectures from patterns or sets of examples and nonexamples; and Overview: Materials: Students will perform simple experiments and investigate the relationship between the experimental probabilities of a simple event and its complement. Group data will be collected and students will compare the group data with the theoretical probability of the event. Number cubes/dice or a six section spinner (enough for every student to have one) Complements Come Easily Handout Group Data Collection Sheet Grouping: Individual and small groups (3 or 4) Time: One 45 minute class Lesson: Procedures 1. To set up this lesson, ask students to define the following: Experiment Event Sample space Notes Students can define by example, such as an experiment is rolling a number cube, drawing a card from a standard deck of 52 cards, or spinning a spinner. An experiment is an action where the results can be recorded. An event is something we are looking for, such as a roll of 5, or the ace of spades, or red on a spinner. Complements Come Easily! Page 1

2 Procedures 2. Distribute the Complements Come Easily! handout. Explain that each student will collect data, then combine that data with the data from the rest of the group, and finally, all group data will be collected to make up class data. Notes A sample space is a list of all possible outcomes. A sample space can also be shown in a tree diagram. (A tree diagram is most helpful when the outcomes are not equally likely to occur.) Make sure students understand that they need to keep a record of each roll. Each student will report to the small group the number of 1 s, 2 s, 3 s, etc. It is important that students predict first what they think the probability of rolling a 3 is. 3. Walk around the room monitoring the progress of the students. As each student finishes, have him/her report data to the group recorder. When a group has finished, have the group recorder put the data on a transparency of the Group Data Collection Sheet. 4. When all group data has been collected, have each student record three different probabilities. 1. the probability of rolling a three using his/her own personal data. 2. the probability of rolling a three using their group data. 3. the probability of rolling a three using the class data. 5. We have calculated the experimental probability of rolling a three on a number cube using three different groups of data. Actually we could compute the probability of rolling a three mathematically. (Theoretical probability) If is often helpful to have students change these probabilities to decimals so that the values can be compared easily. What is the sample space for the event Roll a three? {1, 2, 3, 4, 5, 6} Is it more likely to roll one number than another? No (This addresses the idea of equally likely events.) How many ways can we roll a Complements Come Easily! Page 2

3 Procedures three? One Notes How many different numbers can be rolled? Six So the Theoretical Probability of rolling a three is 1 6. What is the decimal approximation of 1 6? Which of the probabilities that you computed is closest to one-sixth? The class data should be closest in almost every situation. This is the theory of large numbers. The more trials of an experiment, the closer to the theoretical probability the experimental probability will be. This is a great place to use technology to extend the number of rolls to 1000 or greater. This is a screen shot from the Probability simulator on the TI-83. Number of rolls, 100. Note that even with just 100 rolls the bars are close to the same height. 7. Now have students return to their own data and consider the event B, the roll is not a three. Again, have students find the probability of Event B using the three different sets of data. What do you notice about the probability of Event B? Answers will vary. Probability of event A + probability of event B = 1. Complements Come Easily! Page 3

4 Procedures (their own, their group s, and the class data) Notes Subtract Probability of Event A from 1 to get Probability of Event B. If students need more practice with this, there is data for all six possible rolls. Other events that could be explored include: Event C: roll is an even number. Complement: roll is not an even number. Event D: roll is a prime number. Complement: roll is not a prime number. Event E: roll is a square number. Complement: roll is not a square number. 8. Event A and Event B are complementary events. 9. Ask students to brainstorm a variety of experiments and identify events for those experiments and complementary events. Have students add this term to their mathematics vocabulary list. Encourage students to use spinners, playing cards, or colored counters for experiments. 10. Reflection: Have students define each of the following and give an example. Experiment Event Complementary Event Sample space Experimental Probability Theoretical Probability Homework: Design an experiment, and identify an event. Conduct the experiment and compute both the experimental and theoretical probability for the event and its complement. The experiment cannot be roll a standard number cube. Complements Come Easily! Page 4

5 Assessment: The reflection serves as an assessment for this lesson. Complements Come Easily! Page 5

6 Complements Come Easily! Consider the experiment: Roll a number cube. What are the possible outcomes? (sample space) If you roll the number cube 30 times, predict how many times you will roll a 3. (Event A: roll of a number cube is 3) Write your prediction as a fraction; number of 3 s predicted total number of rolls Now conduct the experiment by rolling a number cube 30 times, making a record of what is rolled each time. Based on the results of your 30 trials, write the probability of rolling a 3; number of 3 s rolled total number of rolls Compare the experimental probability of rolling a three with your prediction above. What do you notice? Now, combine your data with the data from the other members of your group. P(Event A group data) = Compare the experimental probability of rolling a three with your prediction above. What do you notice? Record your group data on the Group Data Collection Sheet. The group will need to report the number of 1 s, 2 s, etc. rolled and the total number of rolls. P(Event A class data) = Now consider Event B: the roll is not a three. Using your personal data, what is the experimental probability of Event B? Using your group data, what is the experimental probability of Event B? Complements Come Easily! Page 6

7 How did you find this probability? Is there another way to find this probability? If so, explain. Event A (roll is a 3) Event B (roll is not a 3) Fraction Decimal Fraction Decimal Prediction XXXXXXXX XXXXXXXX Individual Data Group Data Class Data Complements Come Easily! Page 7

8 Group Data Collection Sheet Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group 10 Total Complements Come Easily! Page 8

9 Complements Come Easily! Possible answers Consider the experiment: Roll a number cube. What are the possible outcomes? (sample space) {1, 2, 3, 4, 5, 6} If you roll the number cube 30 times, predict how many times you will roll a 3. 5 or 6_ (Event A: roll of a number cube is 3) Write your prediction as a fraction; number of 3 s predicted total number of rolls 5 30 Now conduct the experiment by rolling a number cube 30 times, making a record of what is rolled each time. Roll Tally Frequency Based on the results of your 30 trials, write the probability of rolling a 3; number of 3 s rolled total number of rolls 6 30 Compare the experimental probability of rolling a three with your prediction above. What do you notice? answers will vary Complements Come Easily! Page 9

10 Now, combine your data with the data from the other members of your group. P(Event A group data) = Compare the experimental probability of rolling a three with your prediction above. What do you notice? answers will vary Record your group data on the Group Data Collection Sheet. The group will need to report the number of 1 s, 2 s, etc. rolled and the total number of rolls. P(Event A class data) = Now consider Event B= the roll is not a three. Using your personal data, what is the experimental probability of Event B? Using your group data what is the experimental probability of Event B? How did you find this probability? Is there another way to find this probability? If so, explain. Prediction Event A (roll is a 3) Event B (roll is not a 3) Fraction Decimal Fraction Decimal XXXXXXXX XXXXXXXX Individual Data Group Data Class Data Complements Come Easily! Page 10

11 Group Data Collection Sheet Sample Data Group Group Group Group Group Group Group Group 8 Group 9 Group 10 Total Total Rolls 630 Complements Come Easily! Page 11