Intent In this activity, students determine some simple two-step probabilities and create related probability questions of their own. It is an opportunity to practice the mathematics developed in recent activities.

Mathematics This activity concludes a sequence designed to solidify the basic definition of probability and introduce area models as a tool for understanding complex probability problems.

Progression Have students do this activity individually, likely as homework. Small-group and class discussions should provide an opportunity to share answers, methods, and reasoning and confirm student confidence in being able to solve similar problems using an area model.

Approximate Time 30 minutes for activity (at home or in class) 15 minutes to share ideas

Classroom Organization Individuals, followed by small-group and whole-class discussions

Doing the Activity This activity will need little or no introduction.

Discussing and Debriefing the Activity Students may have found the answer to Question 1 by adding up the number of cases giving each possible even result (or each odd result). They would find one way to get 2, three ways to get 4, and so on, for a total of 18 ways to get an even result. The probability of an even sum is thus 1836 size 12{ { {18} over {36} } } {} , which simplifies to 12 size 12{ { {1} over {2} } } {} . Similarly, students might add the appropriate probabilities, in this case P(2) + P(4) + P(6) + P(8) + P(10) + P(12). So, the probability of an even sum is 136+336+536+536+336+136=1836=12 size 12{ { {1} over {"36"} } + { {3} over {"36"} } + { {5} over {"36"} } + { {5} over {"36"} } + { {3} over {"36"} } + { {1} over {"36"} } = { {"18"} over {"36"} } = { {1} over {2} } } {}. This is a good opportunity to point out that if students get a simple answer such as 12 size 12{ { {1} over {2} } } {} after performing some complicated arithmetic, they might look for an easier way to find the answer. In this case, they might notice that half the sums in each row of the rug diagram are even. For Question 3, students will have to go through an analysis similar to that done for The Theory of Two-Dice Sums. There are more possibilities for two-dice products than for two-dice sums. The most common products are 6 and 12, each with a probability of 19 size 12{ { {1} over {9} } } {}. For Question 4, students should see that even products are more common. You may want to ask, Why are odds so much less common as products than as sums? Someone may be able to articulate that a product is odd only when both factors are odd, which happens only one-fourth of the time when rolling dice.

Key Question Why are odds so much less common as products than as sums?

Supplemental Activities Different Dice (reinforcement) is similar to Two-Dice Sums and Products and brings out the notion that the sum of all the probabilities in a given situation must equal 1. You might use it with students who need more practice with ideas like those in that assignment. Three-Dice Sums (extension or reinforcement) explores the same ideas in a more complex case.