Intent

This activity poses two more challenging expected value questions. Students will continue to develop methods to solve such problems and work through the meaning of expected value in various contexts.

Mathematics

Students find probabilities and expected values for a variety of situations. These problems become more challenging as the players in the “game” pay each other, and the context of a fair game is brought forth once again. Students continue to determine expected value through the “large number of trials” approach, and the large number has likely become a “convenient” number. Again, student reasoning and student-generated algorithms are the goal, rather than the derivation of a formula or method based on multiplying probabilities and points.

Progression

Students will work on this activity in small groups. Some students may draw area models to help solve the problems; others may be able to work through the tasks without using a visual representation by considering a large number of trials. Prior to moving onto Question 2, a more difficult situation, students should share ideas about how to make the game in Question 1 fair.

Approximate Time

35 minutes

Classroom Organization

Small groups, followed by whole-class discussion

Materials

The tasks in this activity can be challenging. Have paper clips (for making spinners) and decks of cards on hand in case students would like to use them to help them think about the problems.

Doing the Activity

You might engage students in the activity by introducing the first situation using an overhead spinner and asking students to guess who would come out ahead.

Tell students they are to work in their groups to solve the two problems in the activity, but that everyone must record and justify solutions. This may be a good opportunity to collect student work and assess understanding of the concept of expected value.

As students work, listen for their ability to explain their methods to one another. Prepare to ask questions during the class discussion based upon what you’ve heard.

Some students may neglect to consider the fact that Al and Betty pay each other. If a group fails to pick up on this, you might ask, Where do the players get their money? What impact will that have on how much money they are left with after playing a large number of games?

Groups who seem confident can be challenged to extend Question 1b with the question, What other payment systems will make the game fair? and even, Can you describe all the payment systems that create a fair game?

When groups have finished working on Question 1, bring everyone together for a class discussion.

Discussing and Debriefing the Activity

Have groups compare methods for tackling Question 1. Bring out both an area model approach and a “large number of trials” method. If you used an area model, how did you divide the area? If you used a large number of trials, how many did you use?

There are many ways to make the spinner game fair. One of the simplest is to change Al’s payoff to $1.20, leaving Betty’s payoff at 30¢.

Let groups return to Question 2. If time allows, discuss this question as well, which has some tricky aspects. For instance, since the charity receives a penny from each player if neither a jack nor a heart is drawn, the charity will get 2¢ each time it wins. Moreover, if the jack of hearts is drawn, both players have to pay and get paid, making a net gain of 12¢ for Ari. It might help to draw separate area models for Ari, Brenna, and the charity.

Key Questions

Where do the players get their money? What impact will that have on how much money they are left with after playing a large number of games?

If you used an area model, how did you divide the area? If you used a large number of trials, how many did you use?

What other payment systems will make the game fair?

Can you describe all payment systems that create a fair game?

Supplemental Activity

A Fair Dice Game? (reinforcement or extension) requires students to apply their knowledge of two-dice sums as well as the concept of a fair game, so it is a bit more difficult than either A Fair Rug Game? or the supplemental activity Fair Spinners.