Lesson 6Estimating Probabilities Using Simulation

Learning Goal

Let’s simulate real-world situations.

Learning Targets

  • I can simulate a real-world situation using a simple experiment that reflects the probability of the actual event.

Lesson Terms

  • chance experiment
  • event
  • outcome
  • probability
  • random
  • sample space
  • simulation

Warm Up: Which One Doesn’t Belong: Spinners

Problem 1

Which spinner doesn’t belong?

  1. Circular spinner that is 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 “R.”
  2. Circular spinner that is 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."
  3. Circular spinner that is divided into four unequal parts. The first part is yellow and labeled "Y." 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 red and labeled “R.” It is approximately between five-tenths and six-tenths of the spinner. The pointer is in the part labeled "R."
  4. Circular spinner that is 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 red and labeled "R." It is approximately one third of the spinner. The last part is yellow and labeled "Y." It is approximately one fourth of the spinner. The pointer is in the part labeled "R."

Activity 1: Diego’s Walk

Problem 1

  1. 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 had 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.

    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
    had to wait

    fraction
    of days Diego
    had to wait

    1

    no

    2

    yes

    3

    yes

    4

    5

    6

    7

    8

    9

    10

    11

    12

    13

    14

    15

  2. Based on the data you have collected, do you think the fraction of days Diego has to wait after the 16th day will be closer to 0.9 or 0.7? Explain or show your reasoning.

  3. 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
    had to wait

    fraction
    of days Diego
    had to wait

    16

    17

    18

    19

    20

    21

    22

    23

    24

    25

  4. What do you notice about the graph?

  5. Based on the graph, estimate the probability that Diego will have to wait more than 1 minute to cross the crosswalk.

Are you ready for more?

Let’s look at why the values tend to not change much after doing the simulation many times. 

Problem 1

After doing the simulation 4 times, a group finds that Diego had to wait 3 times. What is an estimate for the probability Diego has to wait based on these results?

  1. If this group does the simulation 1 more time, what are the two possible outcomes for the fifth simulation?

  2. For each possibility, estimate the probability Diego has to wait.

  3. What are the differences between the possible estimates after 5 simulations and the estimate after 4 simulations?

Problem 2

After doing the simulation 4 times, a group finds that Diego had to wait 3 times. What is an estimate for the probability Diego has to wait based on these results?

  1. If this group does the simulation 1 more time, what are the two possible outcomes for the twenty-first simulation?

  2. For each possibility, estimate the probability Diego has to wait.

  3. What are the differences between the possible estimates after 21 simulations and the estimate after 20 simulations?

Problem 3

Use these results to explain why a single result after many simulations does not affect the estimate as much as a single result after only a few simulations.

Activity 2: Designing Experiments

Problem 1

For each situation, describe a chance experiment that would fairly represent it.

  1. Six people are going out to lunch together. One of them will be selected at random to choose which restaurant to go to. Who gets to choose?

  2. After a robot stands up, it is equally likely to step forward with its left foot or its right foot. Which foot will it use for its first step?

  3. In a computer game, there are three tunnels. Each time the level loads, the computer randomly selects one of the tunnels to lead to the castle. Which tunnel is it?

  4. Your school is taking 4 buses of students on a field trip. Will you be assigned to the same bus that your math teacher is riding on?

Lesson Summary

Sometimes it is easier to estimate a probability by doing a simulation. A simulation is an experiment that approximates a situation in the real world. Simulations are useful when it is hard or time-consuming to gather enough information to estimate the probability of some event.

For example, imagine Andre has to transfer from one bus to another on the way to his music lesson. Most of the time he makes the transfer just fine, but sometimes the first bus is late and he misses the second bus. We could set up a simulation with slips of paper in a bag. Each paper is marked with a time when the first bus arrives at the transfer point. We select slips at random from the bag. After many trials, we calculate the fraction of the times that he missed the bus to estimate the probability that he will miss the bus on a given day.