Intent

In this activity, students complete a theoretical analysis of the one-and-one situation. In the process, they expand their use of area models to determine the theoretical probabilities in a multistage event and to compute expected value.

Mathematics

The theoretical analysis of the one-and-one situation continues to develop students’ understanding of expected value—not only its computation, but also its meaning as an average over the long run, as opposed to what is most likely to occur. Students also continue their use of the area model as a tool to figure the probabilities of a multistage event.

This understanding of probability and expected value will be used in the final analyses of the games of Little Pig and Pig itself.

Progression

Students will begin this activity in groups. When some groups have completed the activity, the class will come together to share ideas about techniques and the answers they yield.

Approximate Time

40 minutes

Classroom Organization

Small groups, followed by whole-class discussion

Doing the Activity

Students have completed an experimental analysis of the one-and-one situation. Now they will work in groups to develop a rug diagram analysis of Terry’s expected value for each one-on-one situation. (You may want to begin using the term area model in place of rug diagram.)

You may need to help groups get started. One useful and familiar way to begin this analysis is to consider a large, convenient number of cases. The shooting probabilities are reported as percents, which can suggest imagining 100 cases. Consider a large number of one-and-one situations, such as 100. In those 100, Terry would make 60 of her first shots. Can you show that in a rug diagram? A diagram showing the outcomes after the first shot for 100 cases might look like this.

Figure 1

What does this rug diagram mean so far? What do you need to do next?

Some students may find it useful to work with grid paper, using a 10-by-10 section to represent 100 shots.

Figure 2

If they do use grid paper, be aware that later problems may not lend themselves so nicely to whole-number solutions, so students should also work with more schematic diagrams that encourage the computation of multistep probabilities by figuring area through multiplication and addition rather than by counting squares.

Some students may not know how to proceed, especially if they are thinking about each individual one-and-one situation. If so, you might ask what would happen in the cases in which Terry makes her first shot. When Terry gets the opportunity to attempt a second shot, what portion of the time will she make this shot? How will you show this in your area model? The result might look something like this.

Figure 3

Students might develop and label this final area model in various ways. Here is one possibility.

Figure 4

As you circulate, you may want to identify one or two groups whose members seem to have a clear understanding of the process and ask them to prepare presentations. In this case, presenters may be more able to share their thinking about how they designed their area models if they begin with unmarked rugs and talk through each decision in their analyses.

Discussing and Debriefing the Activity

Ask one or two groups to present their work by demonstrating each step of their analyses and how it gets incorporated into their area models. Emphasize the arithmetic of finding the portion of the total area for the various sections. For example, if students use a diagram like the one shown above, they will need to figure out that the “2 points” section represents 36% of the total area, because it is 60% of 60%.

As groups present their diagrams, they should note that each section contains both

If the diagram is drawn on a 10-by-10 grid, each box would represent a single one-and-one situation, and students would be able to find areas by counting boxes.

By considering both the number of cases and the point value for each section, students should come up with an analysis that is something like this, which is based on 100 cases:

Thus 100 cases give a total of

So, the average number of points per one-and-one situation is 0.96. This analysis also confirms that a score of 0 points is most likely (40% of the time), a score of 2 points is next most likely (36% of the time), and a score of 1 is least likely (24% of the time).

This analysis can be done without the use of an area model by simply analyzing what might happen in a large number of cases. But the combination of the geometric and arithmetic perspectives is generally helpful for students, and the technique of subdividing cases or area portions will be useful in analyzing the game of Pig.

This is a good time to reemphasize that the average result is the same no matter how many cases are considered. For instance, considering a total of 1000 one-and-one situations would give a total of 960 points, resulting in the same average of 0.96 point per one-and-one situation. You might mention that expected average might be a better term for this concept, but that expected value has become standard.

The rug diagram can help students confirm that the most likely score (which is 0) is not the expected value (which is very close to 1). The diagram might also provide insight into the misguided intuition that these values are the same, because some students can see in the diagram something “1-like” about the problem. Ask, What’s the difference betweenexpected valueandmost likely outcome? Stress that these two ideas are different, although often confused.

You might ask students to compare the one-and-one situation with other expected value problems they have looked at. One contrast to bring out is that in the one-and-one situation, the outcome did not happen all at once. In 60% of the cases, there was a second event to consider.

To keep students aware of the motivating unit problem, you might ask how this sort of analysis applies to the game of Pig. What connections do you see in the theoretical analysis for one-and-one and our ideas about the game of Pig? Because the number of rolls in Pig may consist of one roll or many, students can expect a “subdivided rug” to show up when they analyze that game.

Key Questions

Consider a large number of one-and-one situations, such as 100. In those 100, Terry would make 60 of her first shots. Can you show that in the area model?

When Terry gets the opportunity to attempt a second shot, what portion of the time will she make this shot? How will you show this in a rug diagram?

What’s the difference between expected value and most likely outcome ?

How does the one-and-one situation compare with other expected value problems you have looked at?

What connections do you see in the theoretical analysis for one-and-one and our ideas about the game of Pig?

Supplemental Activities

Free Throw Sammy (reinforcement) is a follow-up to The Theory of One-and-One. It could be used as a homework assignment instead of Streak-Shooting Shelly if you think that assignment is too difficult. This activity is simpler because Sammy’s probability of making a shot does not change.

Which One When? (extension) brings out the idea that knowing the expected value for a situation does not always give all the information needed to make a decision.