Intent

Students once again consider strategy while examining a game involving probabilities for the sum of two dice. This activity leads to the theoretical analysis of these probabilities.

Mathematics

The counters game provides another context for examining issues of strategy. The game is too complex for students to find the actual probability of success for each strategy. Through experimental analysis, however, students can build intuition about what strategies seem to work best. The context sets the stage for developing tools to determine theoretical probabilities for two-step situations.

Progression

Students first play this game in a group to get a sense of what might be a good or a poor strategy. Then groups come together to play a game as a class. Students will quickly recognize that some two-dice sums occur more frequently than others, motivating the need to derive the theoretical probability for each sum. This theoretical work is done in The Theory of Two-Dice Sums.

Approximate Time

35 minutes

Classroom Organization

Small groups, followed by whole-class discussion

Materials

Dice

Chips (11 for each student)

Doing the Activity

Have students read the instructions for the game. Clarify that with each number rolled, every player removes one and only one counter from that numbered space. Each individual will need a strip of paper for a game board and 11 counters, and each group will need a pair of dice.

Before students begin, you might want to encourage them to first think of a strategy and to place their counters according to it. Allow time for groups to play several rounds and test various strategies. Suggest to them that learning the game and keeping accurate data to test how well each strategy works is more important that winning a particular round.

After groups have played a few games and students have written and thought about strategies, bring the class together for the competition proposed in Question 4. Each group will use a single game board and collectively decide on a strategy for placing counters. Have groups write down their strategies and place their counters accordingly. When groups are ready, begin the game by rolling a pair of dice.

Discussing and Debriefing the Activity

Ask each group to briefly explain its strategy for placing counters. Here are two additional questions you might ask:

Did some two-dice sums come up more often than others? Why?

Is it best to put all your counters on the two-dice sum that you think will come up most often?

Most students will see that some two-dice sums (such as 6, 7, and 8) come up more frequently and that others (especially 2, 3, 11, and 12) come up rarely. Their intuition will probably suggest that it makes sense to concentrate counters in the boxes with sums that come up more often.

A major goal of the activity is to get students interested in the following question, which you should raise if no one else does.

How might you find the theoretical probability of each two-dice sum?