Unit 8, Lesson 6
Estimating Probabilities Using Simulation
Grade 7
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This Warm-up prompts students to compare four images of spinners. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
Arrange students in groups of 2–4. Display the image for all to see. Ask students to indicate when they have noticed which image does not belong and can explain why. Give students 2 minutes of quiet think time and then time to share their thinking with their group. After everyone has conferred in groups, ask the group to offer at least one reason each image doesn’t belong. Follow with a whole-class discussion.
Which three go together? Why do they go together?
Divided into four equal parts: a blue section, labeled “B,” a green section, labeled “G,” a red section, labeled “R,” and a yellow section labeled "Y." The pointer is in the part labeled “Y.”
Divided into three unequal parts. The first part is blue and labeled "B." It is approximately one third of the spinner. The next part is green and labeled “G.” It is approximately one fourth of the spinner. The last part is yellow and is labeled “Y.” It is approximately between four tenths and five tenths of the spinner. The pointer is in the part labeled "Y."
Divided into four unequal parts. The first part is red and labeled "R." It is approximately one eighth of the spinner. The next part is blue and labeled "B." It is approximately one twelfth of the spinner. The third part is green and labeled “G.” It is approximately one fourth of the spinner. The last part is yellow and labeled “Y.” It is approximately between five-tenths and six-tenths of the spinner. The pointer is in the part labeled "Y."
Divided into four unequal parts. The first part is blue and labeled "B." It is approximately one fourth of the spinner. The next part is green and labeled "G." It is approximately one sixth of the spinner. The next part is yellow and labeled "Y." It is approximately one third of the spinner. The last part is red and labeled "R." It is approximately one fourth of the spinner. The pointer is in the part labeled "Y."
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Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure the reasons given are correct.
During the discussion, prompt students to explain the meaning of any terminology they use, such as “outcome,” “region,” or “sample space,” and to clarify their reasoning as needed. Consider asking:
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To Copy (from Blackline Masters)
Diego's Walk Cards
In this activity, students estimate the probability of a real-world event by simulating the experience with a chance experiment. They must make assumptions to simplify the situation and represent it as a simulation as well as analyze the results of the simulation to make a prediction (MP4). Students see that multiple simulation methods can result in similar estimates for the probability of the actual event.
The bag of papers and spinner are designed to have a probability of 0.7 to wait more than 1 minute. The number cubes have a probability of approximately 0.72 to wait more than 1 minute. To the extent that the students are estimating the probabilities, these are close enough to give similar results.
Arrange students in groups of 3. Prepare each group with supplies for 1 type of simulation: choosing a slip from a bag, spinning a spinner, or rolling 2 number cubes. The supplies for these simulations include:
Set up the simulation by telling the students, “Diego must cross a busy intersection at a crosswalk on his way to school. Some days he is able to cross immediately or wait only a short while. Other days, he must wait for more than 1 minute for the signal to indicate he may cross the street. We will simulate his luck at this intersection using different methods and estimate his probability of waiting more than 1 minute.”
Tell students that a simulation is a chance experiment that can be used to estimate the probability of an event. They will use the materials they are given to simulate Diego’s walk.
Give students 15 minutes for group work, and follow with a whole-class discussion.
Your teacher will give your group the supplies for one of the three different simulations. Follow these instructions to simulate 15 days of Diego’s walk. The first 3 days have been done for you.
Simulate one day:
If your group gets a bag of papers, reach into the bag, and select one paper without looking inside.
If your group gets a spinner, spin the spinner, and see where it stops.
If your group gets two number cubes, roll both cubes, and add the numbers that land face up. A sum of 2–8 means Diego has to wait.
Record in the table whether or not Diego has to wait more than 1 minute.
Calculate the total number of days and the cumulative fraction of days that Diego has had to wait so far.
On the graph, plot the number of days and the fraction that Diego has had to wait. Connect each point by a line.
If your group has the bag of papers, put the paper back into the bag, and shake the bag to mix up the papers.
Pass the supplies to the next person in the group and they will repeat the steps to simulate the next day.
A graph of two connected line segments on a coordinate grid with the origin marked “O.” The horizontal axis is labeled “day”, with the numbers 0 through 24, in increments of 2, indicated. There are vertical grid lines midway between. The vertical axis is labeled “fraction of days Diego had to wait” with the numbers 0 point 1 through 1, in increments of 0 point 1, indicated. The first line segment begins at the point with coordinates 1 comma 0 and moves upward and to the right, ending at the point with coordinates 2 comma 0 point 5. The second line segment begins where the first line ends, and moves slightly upward and to the right, ending at the point with coordinates 3 comma 0 point 6 7.
| day | Does Diego have to wait more than 1 minute? |
total number of days Diego has had to wait |
fraction of days Diego has had to wait |
|---|---|---|---|
| 1 | no | 0 |
|
| 2 | yes | 1 |
|
| 3 | yes | 2 |
|
| 4 | |||
| 5 | |||
| 6 | |||
| 7 | |||
| 8 | |||
| 9 | |||
| 10 | |||
| 11 | |||
| 12 | |||
| 13 | |||
| 14 | |||
| 15 |
Based on the data you have collected, do you think the fraction of days Diego has had to wait after the 16th day will be closer to 0.9 or 0.7? Explain or show your reasoning.
Continue the simulation for 10 more days. Record your results in this table and on the graph from earlier.
| day | Does Diego have to wait more than 1 minute? |
total number of days Diego has had to wait |
fraction of days Diego has had to wait |
|---|---|---|---|
| 16 | |||
| 17 | |||
| 18 | |||
| 19 | |||
| 20 | |||
| 21 | |||
| 22 | |||
| 23 | |||
| 24 | |||
| 25 |
The purpose of this discussion is for students to understand why simulations are useful in place of actual experiments.
Select at least 1 group for each of the simulation methods to display the materials they used to run their simulation and explain the steps involved in using their materials.
Ask students, “Why do you think these simulations are more useful than actually doing the experiment many times?” (It would take a lot of time and work for Diego to walk to school more than usual, but it is easy to do the simulation many times quickly.)
Select students to share what they noticed about the graph of the fraction of days Diego had to wait as the simulated days went on.
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Students may think that the number of outcomes in the sample space must be the same in the simulation as in the real-life situation. Ask students how we could use the results from the roll of a standard number cube to represent a situation with only two equally likely outcomes. (by making use of some extra options to count as “roll again”)
Building Toward
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