Intent

Students analyze another probabilistic situation involving a payoff by considering a large number of trials. In the discussion of the activity, they are introduced to the term expected value.

Mathematics

The “large number of trials” approach used in Spinner Give and Take and Mia’s Cards gives the concept of expected value a solid intuitive foundation. The term expected value is introduced in the discussion of Mia’s Cards, with the acknowledgment that this is simply a formal name for an idea that students have already encountered in the context of examining averages in the long run.

Over the course of the activities in In the Long Run, students apply the concept of expected value in increasingly complex situations. Near the end of the unit, they will use expected value to evaluate strategies for a game similar to Pig, in preparation for justifying the best strategy for Pig.

Progression

Students work individually on the activity and then come together to share methods and ideas. The teacher then introduces the term expected value for the concept of computing an average per turn over the long run. Students will connect the concept to previous work in the unit and to the unit problem: finding the strategy for the game of Pig that gives the highest expected value.

Approximate Time

5 minutes for introduction

20 minutes for activity (at home or in class)

30 minutes for discussion

Classroom Organization

Individuals, then groups, followed by whole-class discussion

Doing the Activity

This activity is similar to the earlier spinner and rug games, though students may find it more difficult, as the situation is more complex. If assigning the activity as homework, make sure students know the number of cards and suits in a typical deck.

Discussing and Debriefing the Activity

Have students share their results in groups. As they are doing so, you might give some groups transparencies, assign one or two groups each to Questions 1 and 2, and have other groups choose a game that one of their members invented for Question 3.

Presentations of solutions to Questions 1 and 2 can help students feel comfortable working with a large number of games, finding the total number of points, and then finding the average by dividing by the number of games.

Presentations that use different numbers of games will help dramatize the point that the average will be the same. For example, in Question 1, using 100 games gives Mia about 25 hearts, for 250 points, and about 75 cards from the other suits, for 375 points. Using 500 games gives her about 1250 points from hearts and 1875 points from the other suits, for a total of 3127 points. Each total gives an average of about 6.25 points for each card picked.

Introduce the term expected value for “the average amount gained (or lost) per turn in the long run.” Ask students to restate their results from Questions 1 and 2 using this term. For instance, for Question 1, students might say that Mia has an expected value per turn of 6.25 points. (For emphasis, it’s a good idea to use a phrase like per game or per turn whenever you use the term expected value.)

Help students see that expected value is an average in the long run, not what one expects on any particular trial. Ask, So Mia will get 6.25 points each time she draws a card? You may want to emphasize that expected value is not a new concept, but simply a concise name for an idea students have already worked with. You can point out that the use of such terminology makes it easier to state complex ideas. [link to math maps]

Students can gain practice with the terminology by rephrasing their results from Waiting for a Double and from Question 3 of this activity. Ask, Can you rephrase other problems you’ve seen in terms of expected value? Students should notice that their work in Waiting for a Double gave an experimental estimate of the expected value for the number of rolls needed to get a double.

If students are having difficulty, use some of the games they made up for Question 3 to solidify the idea. Even if they seem comfortable with the concept of expected value, you might ask groups to share some of their games and have the class figure the expected value of each game.

If students haven’t already made the connection between the concept of expected value and the game of Pig, you might ask, How can you restate the unit problem in terms of expected value? Help students to realize that they are looking for the Pig strategy that gives the highest expected value. Edit or add to the posted unit goal to reflect this use of the term.