The Theory of Two-Dice Sums
Intent
Students extend their use of area models to find the theoretical probability distribution for two-dice sums. In the process, they develop a tool for determining theoretical probabilities in a multistep situation.
Mathematics
The notion of a probability distribution is introduced through the use of frequency bar graphs. The term rug diagram is not standard among mathematicians. The transition to using the more common term area model can be gradual.
Progression
In this activity, students begin to design rug diagrams to help them determine theoretical probabilities. After some work in groups to develop a theoretical model, the class clarifies the model and makes connections between the theoretical probabilities and the data collected in Rollin’, Rollin’, Rollin’.
Over the course of the unit, students will work with probabilities associated with increasingly complex situations. Later they will encounter situations in which the outcomes are not equally likely and use two-dimensional diagrams to determine the probabilities.
Approximate Time
20 minutes for activity
30 minutes for discussion
Classroom Organization
Groups, followed by whole-class discussion
Materials
Dice in two colors
Doing the Activity
Tell students that they will now develop a tool to determine the theoretical probability for each possible two-dice sum. You can start them thinking about the task by asking, How would you draw a rug diagram to show what could happen after rolling one die?
Give each group a red die and a white die (or two other colors) to begin working on the task. Using dice of two colors can help students understand, for example, that there are two ways to roll 3 and 5 (red 3 and white 5 or red 5 and white 3), while there is only one way to roll two 4s.
You will need to decide whether students are making progress on their own or whether it would be more productive to bring them together to share ideas. (You might choose to have students interrupt the work on this activity in order to complete Money, Money, Money, as that activity has only three possible outcomes, and then return to their work on this activity.)
Ask groups that are stuck what could happen after rolling 1 on the first die and how that could be shown in a rug diagram. Can you think of a way to show the possible results on the second die, if you rolled 1 on the first die? This question may encourage students to subdivide each of the six initial outcomes.
Discussing and Debriefing the Activity
Students may derive a diagram something like this.
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To emphasize the “equally likely” aspect of the problem, ask, Which column is the most likely to occur? Which row? Which square is the most likely to occur when rolling two dice? Why are they all equally likely?
Students should be able to articulate the idea that each square is equally likely, with a probability of
You can also ask for the probability of rolling individual combinations, such as 3 on the red die and 6 on the white die. Students should see that each combination has a probability of
You might also ask such questions as, Which part of the diagram represents a roll of 4 on the white die?
Next, ask students to identify the portion of the diagram that represents a roll totaling 6. The probability of getting 6 as the two-dice sum is
By the end of the discussion, students should realize there are 36 equally likely outcomes for a pair of dice and be able to find the probability for each of the 11 possible sums (2 through 12).
Here are some additional questions you can pose to strengthen and clarify students’ understanding of the area model:
Why are there two ways of getting a sum of 3, when the only way to get it is with a 1 and a 2? Students should see that “red = 1, white = 2” is a different square from “red = 2, white = 1.”
What is P(even number)?
What is P(multiple of 5)?
Have students compute what the class totals would have been for Rollin’, Rollin’, Rollin’ if the experiments had followed the theoretical probabilities. What would the totals have been if our experiments had followed the theoretical probabilities perfectly? Describe the answer as the theoretical distribution or theoretical probability distribution.
You might have students make a graph of this theoretical distribution and compare it to the graph of their experimental data. They will probably see some general similarity, but they will also see that their results vary somewhat from the theoretical distribution. You might ask, Which is closer to the theoretical distribution—your individual data or the combined class data? Why?
Post the rug diagram for two-dice sums for later use.
Key Questions
How would you draw a rug diagram to show what could happen after rolling one die?
Can you think of a way to show the possible results on the second die, if you rolled 1 on the first die?
Which column is the most likely to occur? Which row?
Which square is the most likely to occur when rolling two dice? Why (or why are all equally likely)?
What is the probability of getting a red 3 and a white 6?
Which part of the diagram represents a roll of 4 on the white die?
How does the diagram represent a sum of 6 on the two dice?
What is P(even number)? What is P(multiple of 5)?
What would the totals have been if our experiments had followed the theoretical probabilities perfectly?
Which was closer to the theoretical distribution—your individual data or the combined class data?
Supplemental Activity
Two-Spinner Sums (extension or reinforcement) is similar The Theory of Two-Dice Sums, but with a slightly different setting. In this set of two tasks, students examine the results of spinning two spinners and compare this work to the results of tossing two dice.
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Last edited by Christine Osborne on Jun 18, 2008 12:12 pm GMT-5.