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In the Long Run

Module by: Interactive Mathematics Program

Intent

Expected value—the average value over the long run—is the focus of the activities in In the Long Run. Students develop ways of computing expected value in a variety of situations. They continue to use area models, and are introduced to tree diagrams, in their analyses of the long-run behavior of probabilistic events.

Mathematics

For the rest of this unit, the notion of in the long run will be emphasized to ground students’ reasoning about the concept and the computation of expected value, essential to analyzing strategies.

Rolling a die is a random process; no one can predict the outcome of any single roll. However, in the long run we can predict with confidence that each number will come up on 1616 size 12{ { {1} over {6} } } {} of the rolls. The sum of the numbers is 21, so in the long run the average number per roll is 216=3.5216=3.5 size 12{ { {"21"} over {6} } =3 "." 5} {}. Expected value is shorthand for average in the long run. The expected value per roll of a die is 3.5.

Expected value ties together theoretical and observed probabilities. The expected value is a theoretical result; in the case of a die, no roll can result in 3.5. But expected value is the best prediction of an observed probability.

One source of observed probabilities is simulations of random processes. Students will use spinners, dice, and playing cards to conduct several simulations in In the Long Run. They also continue to use area models for analyzing multistage situations, and they are introduced to tree diagrams.

Progression

Over the course of In the Long Run, students will examine several games in which the payouts to the players are determined by the results of some random event. Methods for quantifying results are made more formal in these activities, in anticipation of returning to the game of Pig. The final POW of the unit, in which the focus is once again on strategy analysis, is posed early in In the Long Run.

Spinner Give and Take

Pointed Rugs

POW 5: What’s on Back?

Mia’s Cards

A Fair Rug Game?

One-and-One

A Sixty Percent Solution

The Theory of One-and-One

Streak-Shooting Shelly

Spins and Draws

Aunt Zena at the Fair

Simulating Zena