Intent This activity sets the stage for a several-day investigation of a more complex probabilistic situation. Students apply the concept of expected value to this new situation, this time in relation to most likely outcome. They conduct simulations as well.

Mathematics Over the course of the activities in In the Long Run, students apply the concept of expected value in increasingly complex situations. One-and-One presents a two-stage process in which the outcome of the first stage determines whether there is a second stage. Streak-Shooting Shelly also involves two stages, with the probabilities changing in the second stage. The situation in Martian Basketball has up to three stages. One-and-One employs the first focused use of a simulation, in which students model a one-and-one situation in basketball by pulling colored cubes from a bag.

Progression This activity introduces the one-and-one free throw context, which is used in the next several activities. Students first guess how many points they think are most likely for a one-and-one free throw shooter to make and then design a simulation to test their guesses. In the next activity, The Theory of One-and-One, students develop theoretical methods to analyze their initial guesses. Other activities are intermingled for student practice and extension.

Approximate Time 25 minutes

Classroom Organization Whole-class discussion, with some data generated in pairs

Materials Large collection of red and yellow cubes (or similar items in two colors) Small paper bags

Doing the Activity Ask whether anyone can explain what a one-and-one situation is in basketball. A one-and-one occurs in a penalty situation in which one player has committed a foul against another. The player who has been fouled is allowed to take a free throw. If the free throw is unsuccessful, the one-and-one situation is over. If the first shot is successful, the player gets to shoot once more. Each successful shot scores 1 point. The player can thus score a total of 0 points (by missing the first shot), 1 point (by making the first shot but missing the second), or 2 points (by making both shots). Once students understand the one-and-one situation, have them read the problem described in One-and-One, in which Terry has a 60% chance of success on each attempted free throw. Clarify the meaning of this by asking how many shots Terry would be likely to make out of 100, out of 40, out of 15, and so on. Then have students, working individually, consider the question posed in the activity: In a one-and-one situation, how many points is Terry most likely to score: 0, 1, or 2? When they have thought about the question, conduct a class vote, recording the number of students who select each possible score. Tell students that they will now design and conduct an experiment to estimate the probabilities involved in the situation. Introduce the word simulation. Ask, Where have you heard the word simulation before? (One example is flight simulators.) Why do people use simulations? The main idea that should emerge is that a simulation allows us to learn about something when we can’t investigate that thing directly. You might also ask, Have you done any simulations for other mathematics problems in this unit? Students might mention their current work on POW: What’s on Back? Ask students to describe the difference between finding probability using experimental results and using a theoretical analysis. They should be growing more comfortable articulating that an experiment can give them a feel for what the results might be, while a simulation will give only observed probabilities, which can at best approximate theoretical probabilities, even with a large number of trials. Give students the paper bags and cubes and ask, How might you use these materials to set up a simulation to study the question in this activity? Alternatively, you might ask students to design a simulation without prompting them with specific materials. They might suggest using other objects or propose using a random number generator. If students suggest using 60 cubes of one color and 40 of the other, ask whether there is a smaller number of cubes that would work. Some students will identify 3 and 2; others will be more comfortable with 6 and 4. Focus on why the particular combinations they identify are appropriate. Ask, Why is this combination of cubes suited to this problem? Students should be able to articulate that if 60% of the cubes are red, picking a red cube represents making a free throw and picking a yellow cube represents missing a free throw. It’s a good idea to have students try to describe exactly how they will conduct the simulation. For example, they might construct instructions like these: Shake the bag and pull out a cube. If the cube is yellow, the simulation is over. Write 0 for the score. If the cube is red, draw again to complete the simulation. (Return the red cube to the bag and shake before drawing again.) If the second cube is yellow, write 1 for the score. If the second cube is red, write 2 for the score. Conduct a few simulations as a class, and then have students work in pairs. Each pair should simulate the one-and-one situation approximately 20 times, recording each result, to gather lots of data for the class to examine. When they are finished, have each pair tally the number of 0s, 1s, and 2s they got.

Discussing and Debriefing the Activity Compile a class total of the 0s, 1s, and 2s from the simulations. As a class, calculate the percentage of the time each score occurred. If the theoretical probabilities are borne out, the experiment will have produced more 0s than any other number. (However, 2s are a close theoretical second.) Typically, most students will have guessed that a score of 1 is the most likely. This conversation will be continued in the discussion of Theory of One-and-One, when students observe that the expected value per one-and-one situation is very close to 1, even though the score of 1 is actually the least likely result. Ask, Do you think the simulation is a good method for analyzing the situation? Tell students that they will later use an area model as a theoretical model to analyze the situation.

Key Questions In a one-and-one situation, how many points is Terry most likely to score: 0, 1, or 2? Where have you heard the word simulation before? Why do people use simulations? Have you done any simulations for other mathematics problems in this unit? What is the difference between experimental results and theoretical analysis? How would you set up a simulation using these materials? Why is this particular combination of cubes suited to this problem? Do you think the simulation is a good method for analyzing the situation?