Intent

In this POW, students use both experimental and theoretical methods to examine a probability situation. Their success in this activity should be measured in terms of the growth in their understanding of the concepts involved.

Mathematics

This activity provides a context for examining issues of strategy. Although no strategy will always ensure a win in this game, students can determine the strategy that optimizes their chance of winning. Students carry out trials to approximate the experimental probability for each of several strategies and apply techniques developed in the unit to determine a theoretical probability for winning. Students use their intuitive notions about the law of large numbers to justify the reasonableness of relying on experimental probabilities and to confirm the accuracy of the theoretical probabilities.

Progression

Students begin by considering a seemingly simple game in which a player is to guess “what’s on back” of a card pulled from a bag. Asked to determine the strategy with the highest probability of success, students must design experimental and theoretical analyses to test various strategies. The problem is counterintuitive for most people and can be difficult to reconcile in a short period of time. As always, students should work over the course of the week and come together midway to share ideas and discoveries. Students will present their analyses for two given strategies and any others they explore.

You may seek to conclude the discussion by generalizing the possible strategies in order to determine the best strategy. Similar reasoning will be used later in the unit as students return to attempting to find the best strategy for the game of Pig.

Approximate Time

10 minutes for introduction

1 to 3 hours for activity (at home)

10 minutes for follow-up midweek

20 minutes for presentations

Classroom Organization

Individuals, followed by small-group discussions, concluding with whole-class presentations

Materials

Bags

Heavy cardstock

Markers

Doing the Activity

You may also want to talk about the difference between experimental results and results derived from a theoretical model. Although an experiment gives a feel for what the results might be, even a large number of trials does not guarantee an accurate picture of the theoretical probability. If the theoretical analysis gives a result very different from the experimental estimate, this might indicate an error in the theoretical analysis. The following prompt may help initiate this discussion: What are the advantages of experiments in finding probabilities? What are the advantages of theory?

As always, monitor and encourage student work throughout the week. If students are having difficulty starting, you might focus them on the first strategy described in the POW (predicting that the mark on the other side will be different from the mark showing).

It’s good if students realize that they need to do more than a handful of experiments to get an accurate estimate of the probability of success for a particular strategy. Often students feel as though they are “doing the problem” after completing only a few experiments. Conversation and comparison of experimental data may be necessary for them to gain a greater appreciation of the need to do more trials. Ask, How many experiments are you finding you need to do? to encourage reflection on the notion of continuing until the experimental probabilities stabilize.

If needed, remind students that they are also looking for a theoretical analysis of the probability. Ask, How might you develop a theoretical model? In this unit, they have seen theoretical probabilities derived through rug models.

It is a common misconception that if one initially sees X, the probability is .5 that the other side is also X. This idea comes from the fact that there are only two cards containing an X, which suggests that the chosen card is equally likely to be one of these two cards. One of these cards has X on the other side, and one has O on the other side. However, the truth is that if you’re looking at a side containing X, you’re more likely to have drawn the card with an X on both sides than the card with an X on only one side. You might wish to draw attention to this concept after students have worked for a short time. The discussion can highlight the value of using experimental probabilities to test the reasonableness of proposed theoretical probabilities.

As the day for presentations draws near, you might ask for volunteers to present, with each focusing on a different aspect of the problem. For example, one might talk about results with one strategy, another about results with a second strategy, and a third about finding the best strategy.

Discussing and Debriefing the Activity

This POW may have been difficult for many students. Although the content of this activity is related to that of the unit, students certainly don’t need to uncover the best strategy or master the mathematical content to be successful with the activity or the unit. The intent of POWs is to develop problem-solving skills and positive attitudes. Even if students come away from this problem without understanding why a particular strategy is best, they will probably have strengthened their appreciation of several ideas, including

•using an experiment to estimate a probability

•understanding that the accuracy of an experimental estimate will generally depend on how many times the experiment is done

•recognizing that area models are a useful tool for finding theoretical probabilities

During the presentations, focus the discussion on students’ work with the two strategies suggested in the activity. Bring out the idea that an experiment will give a more accurate estimate if it is repeated more often.

In discussing how to determine the best strategy, you may first want to identify what the possible strategies are. Focus students’ attention on strategies that are based on what shows up on the side of the card drawn from the bag. From this perspective, each strategy should look something like:

If you get a card with X showing, then predict ___.

If you get a card with O showing, then predict ___.

As presenters discuss their theoretical analyses, keep in mind that a crucial element of the theoretical analysis of any strategy is determining what the equally likely outcomes are. One approach is to recognize that each card is equally likely to be drawn and, then, that each side of the drawn card is equally likely. This is equivalent to saying that each of the six sides is equally likely to be the initial side viewed. A diagram like the one below can then be very useful in examining strategies. Here, sections above or below each other represent the two sides of a given card.

Figure 1

This diagram can be shaded to indicate cases in which the first strategy, “predict different,” would be successful. The top left box below is unshaded because if you draw the card and observe the face it represents, the other side will be the “same.” The bottom middle box is shaded because if you select that card and observe the O face, the other side will be “different.” Finish the diagram by considering the four remaining boxes.

Figure 2

So, for the “predict different” strategy, Psuccess=13Psuccess=13 size 12{P left ("success" right )= { {1} over {3} } } {}.

A similar analysis for the strategy “always predict X” produces the diagram below and gives Psuccess=12Psuccess=12 size 12{P left ("success" right )= { {1} over {2} } } {}. The strategy “always predict O” also gives Psuccess=12Psuccess=12 size 12{P left ("success" right )= { {1} over {2} } } {}, so either of these strategies is better than the first one.

Figure 3

Finally, the strategy of “predict same” is represented in the next diagram. For this strategy, Psuccess=23Psuccess=23 size 12{P left ("success" right )= { {2} over {3} } } {}.

Thus, “predict same” is the best possible strategy.

Key Questions

In this POW, what does it mean to experiment?

How many experiments might you need to do?

How many experiments are you finding you need to do?

Supplemental Activity

Make a Game (extension) asks students to work in pairs to create games that involve probability and strategy. You may want to provide a clear schedule for the various stages of this activity, such as choosing partners and presenting preliminary plans. Make the grading criteria clear (such as use of probability and the need for strategy, entertainment, and clear instructions), perhaps even giving students a voice in establishing those criteria. You may want to provide class time for students to work on their games and allot a full day toward the end of the unit for them to share and explain their games.