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.

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. 

In this activity, students have the opportunity to design their own simulations that could be used to estimate probabilities of real-life events (MP4). Students attend to precision (MP6) by assigning each possible outcome for the real-life experiment to a corresponding outcome in their simulation in such a way that the pair of outcomes have the same probability. In the discussion following the activity, students are asked to articulate how these simulations could be used to estimate probabilities of certain events.

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”)

The purpose of this discussion is for students to think more deeply about the connections between the real-life experiment and the simulation.

Select groups to share the simulations they designed for each of the situations.

Some questions for discussion:

• “How could a standard number cube be used to simulate the situation with the buses?” (Each bus is assigned a number 1 through 4. If the cube lands on 5 or 6, roll again and do not count it in the simulation.)
• “If one of the buses is numbered with your math teacher's favorite number and you want to increase the probability of that bus being selected, how could you change the simulation to do this?” (Add more of the related outcome. For example, using the standard number cube as in the previous discussion question, the bus with the favorite number could be assigned numbers 4 and 5 while the other buses are still 1 through 3.)
• “Two of the tunnels in the video game lead to a swamp that ends the game. How could you use the simulation to estimate the probability of choosing one of those two tunnels?” (Since all of the tunnels are equally likely to lead to the swamp, it can be assumed that “left” and “right” lead to the swamp. Spin the spinner many times and use the fraction of times it ends on “left” or “right” to estimate the probability of ending the game. It should happen , or about 67%, of the time.)
• “You and a friend are among the people going to lunch. How could you use the simulation you designed to estimate the probability that you or your friend will be the one to choose the restaurant?” (My friend and I will be represented by 1 and 2 on a number cube. Roll the number cube a lot of times and find the fraction of times 1 or 2 appear, then estimate the probability that one of us will be the ones selected.)