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Another Carrier Dilemma

Module by: Interactive Mathematics Program

Intent

Students use a variety of approaches to find the expected value in one more multi-step carrier situation.

Mathematics

Students continue to practice the three methods of calculating expected value (area model, tree diagram, and systematic list) in a multi-step situation.

Progression

This is an additional real-life situation for students to consider before returning to the unit problem.

Approximate Time

20 minutes for activity (at home or in class)

10 minutes for discussion

Classroom Organization

Individuals, followed by whole-class discussion

Doing the Activity

Clarify the point that students must use at least two methods to compute the expected value for the carrier’s weekly earnings in this new version of the problem.

Discussing and Debriefing the Activity

Have at least one student present each method—an area model, a tree diagram, and a list of combinations—with a focus on finding the probability of each possible outcome. Finding the expected value once the probabilities are known should be fairly routine by now, and students should also understand that the expected value will be the same no matter which method they use.

You many want to share with students the area model below, in which all the bills are listed both across the top and down the side, and in which the impossible cases (representing the same bill being drawn twice) are shaded out. The number in each box shows the total amount the carrier earns.

Figure 1
Figure 1 (graphics1.jpg)

Because the 20 possible cases are equally likely, one can determine by counting that the probability of getting $10 is 220220 size 12{ { {"2"} over {"20"} } } {} , the probability of getting $6 is 12201220 size 12{ { {"12"} over {"20"} } } {}, and the probability of getting $2 is 620620 size 12{ { {6} over {"20"} } } {}. From these probabilities, the expected value over 100 weeks can be computed as 1010+606+3021001010+606+302100 size 12{ { {"10" left ("10" right )+"60" left (6 right )+"30" left (2 right )} over {"100"} } } {}, or $5.20 per week.

Key Question

Does one method for determining expected value seem more direct than another?

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