Unit 8, Lesson 5: More Estimating Probabilities

Transcription

1 Unit 8, Lesson 5: More Estimating Probabilities Lesson Goals Explain why a probability estimate based on repeating an experiment many times may not be exactly equal to the actual probability for a given event. Understand that a probability can be estimated by performing the experiment many times. Required Materials snap cubes copies of blackline master paper clips paper bags 5.1: Is it Likely? (5 minutes) Setup: 2 minutes of quiet work time followed by whole-class discussion. 1. If the weather forecast calls for a 20% chance of light rain tomorrow, would you say that it is likely to rain tomorrow? 2. If the probability of a tornado today is, would you say that there will likely be a tornado today? 3. If the probability of snow this week is 0.85, would you say that it is likely to snow this week? 1. Not likely. 2. Not likely. 3. It is likely. Unit 8: Probability and Sampling, Lesson 5: More Estimating Probabilities 1

2 5.2: Making My Head Spin (20 minutes) Setup: Groups of 4. One copy of the blackline master for each group. One pencil and paper clip for each student. Teacher demonstration, 5 minutes of quiet work time, 10 minutes of group work, followed by whole-class discussion. Unit 8: Probability and Sampling, Lesson 5: More Estimating Probabilities 2

3 Your teacher will give you 4 spinners. Make sure each person in your group uses a different spinner. 1. Answers vary. 1. Spin your spinner 10 times, and record your outcomes. 2. Did you get all of the different possible outcomes in your 10 spins? 3. What fraction of your 10 spins landed on 3? 4. Next, share your outcomes with your group, and record their outcomes. a. Outcomes for spinner A: b. Outcomes for spinner B: c. Outcomes for spinner C: d. Outcomes for spinner D: 5. Do any of the spinners have the same sample space? If so, do they have the same probabilities for each number to result? 2. Answers vary. 3. Answers vary. 4. Answers vary. 5. Yes, spinners A and B have the same sample space. They do not have the same probabilities for the numbers, though. For example, spinner A has two 3s out of 12 equal spaces, so the probability of spinning a 3 is. Spinner B only has one 3 out of 12 equal spaces, so the probability of spinning a 3 is. 6. For each spinner, what is the probability that it lands on the number 3? Explain or show your reasoning. 7. For each spinner, what is the probability that it lands on something other than the number 3? Explain or show your reasoning. 8. Noah put spinner D on top of his closed binder and spun it 10 times. It never landed on the number 1. How might you explain why this happened? 6. Spinner A:. Spinner B:. Spinner C:. Spinner D:. The values for spinners A, B, and C are computed by counting the number of 3s on the spinner and Unit 8: Probability and Sampling, Lesson 5: More Estimating Probabilities 3

4 9. Han put spinner C on the floor and spun it 10 times. It never landed on the number 3, so he says that the probability of getting a 3 is 0. How might you explain why this happened? dividing by the number of equal sections on the spinner. For spinner B, the section for 3 is one fourth of the circle. 7. Spinner A:. Spinner B:. Spinner C:. Spinner D:. Each of these values was computed by counting the number of things that were not 3 on the spinner and dividing by the number of sections on the spinner. 8. Since the binder is sloped, gravity may have pulled the spinner so that a 1 would not show up. 9. Han might have been holding the spinner at an angle like Noah or maybe he just did not spin enough times. Since it is possible to spin a 1, the probability should not be 0. Anticipated misconceptions Students may think they need to have Unit 8: Probability and Sampling, Lesson 5: More Estimating Probabilities 4

5 their probability estimates match the computed probability. Remind them of the activity "Due For a Win" to see why we might expect the estimated probability and the calculated probability to be a little different. Are you ready for more? Design a spinner that has a probability of landing on the number 3. Explain how you could precisely draw this spinner. Possible Responses Answers vary. Sample response: First, I would draw a circle with a compass. Then I would divide the circle into 3 equal sections by using a protractor and measuring an angle of since. I would write the number 3 in two of the sections and write the number 1 in the other section. Unit 8: Probability and Sampling, Lesson 5: More Estimating Probabilities 5

6 5.3: How Much Green? (10 minutes) Setup: Groups of 3 4. Distribute bags of snap cubes (3 green cubes and 2 of some other color per bag). 5 minutes of group work followed by whole-class discussion. Your teacher will give you a bag of blocks that are different colors. Do not look into the bag or take out more than 1 block at a time. Repeat these steps until everyone in your group has had 4 turns. Take one block out of the bag and record whether or not it is green. Put the block back into the bag, and shake the bag to mix up the blocks. Pass the bag to the next person in the group. 1. Answers vary. 2. Answers vary. Sample response: Continuing to pick out blocks more times might get a better estimate. 1. What do you think is the probability of taking out a green block from this bag? Explain or show your reasoning. 2. How could you get a better estimate without opening the bag? Anticipated misconceptions Some students may estimate a probability that is different from the fraction of times they draw a green block. ask these students for a reason they chose a different value for their estimate. Lesson Synthesis (5 minutes) If the sample space is unknown, how can you estimate the probability of an event? How can you improve the estimate after a few trials of an experiment? Unit 8: Probability and Sampling, Lesson 5: More Estimating Probabilities 6

7 5.4: The Probability of Spinning B (Cool-down, 5 minutes) Setup: None. Jada, Diego, and Elena each use the same spinner that has four (not necessarily equal sized) sections marked A, B, C, and D. Answers vary. Jada says, "The probability of spinning B is 0.3 because I spun 10 times, and it landed on B 3 times." Diego says, "The probability of spinning B is 20% because I spun 5 times, and it landed on B once." Elena says, "The probability of spinning B is because I spun 7 times, and it landed on B twice." 1. Based on their methods, which probability estimate do you think is the most accurate? Explain your reasoning. 2. Andre measures the spinner and finds that the B section takes up of the circle. Explain why none of the methods match this probability exactly. Unit 8: Probability and Sampling, Lesson 5: More Estimating Probabilities 7