Intent

Students examine another two-step probability problem. This activity can be done either during The Theory of Two-Dice Sums to help clarify the situation and draw out ideas, or after that activity to reinforce the techniques that emerged.

Mathematics

The two situations posed in this activity are two-step probability problems. If students are able to visualize the possible outcomes using an area model, it is likely they will be able to apply the techniques they have acquired to build an area model for The Theory of Two-Dice Sums.

Progression

Have students do this activity individually, likely as homework. During their work or the discussion, students should recognize that the two situations posed in this activity have a two-stage aspect to them that can be represented by the two-dimensional nature of a rug diagram (area model).

Approximate Time

20 minutes for activity (at home or in class)

20 minutes for discussion

Classroom Organization

Individuals, followed by whole-class discussion

Doing the Activity

This activity needs little or no introduction.

Discussing and Debriefing the Activity

For Question 1, students may see that it makes sense to set up the diagram with the result of one coin across the top and the result of the other coin down the side. With this diagram, they can show that the three outcomes Nina describes are not equally likely.

For Question 2, you can move students toward a diagram like the one below. This may not be the most obvious representation for some students, so expect the need for some clarification. You might ask, What could happen if the $1 bill is pulled from the left pocket? How can the column showing the likelihood of drawing $1 be divided to show the three possible bills drawn from the right pocket?

Students will probably see that each of the nine boxes is equally likely, and they can then find the probabilities.

Rather than explicitly discuss Question 3, you may just want to suggest that students incorporate the methods used in this activity into their two-dice sum analyses, looking for ways to represent equally likely outcomes.

Key Questions

What could happen if the $1 bill is pulled from the left pocket? How can the column showing the likelihood of drawing $1 be divided to show the three possible bills drawn from the right pocket?

From a probability perspective, is it the same or different to draw $5 from my right pocket followed by drawing $1 from my left pocket as it is to draw $5 from the right and $1 from the left?