Intent This activity introduces the notion of a fair game and the use of negative numbers for expected value.

Mathematics In this activity, students consider negative values for expected value and what it means for expected value to be zero. Students will use expected value to compare strategies later in the unit, considering which strategy will yield the most points per turn on average over the long run.

Progression After students have completed their work individually, the discussion should first identify each player’s expected value and the fact that one expected value is the opposite of the other. Next, a discussion of a fair game will set the stage for altering the scoring to make the game fair. Identify that when the game is fair, each player’s expected value is now zero. This activity continues to build students’ fluidity with deriving expected value and understanding its meaning. Students should continue to emphasize a “large number of trials” approach.

Approximate Time 5 minutes for introduction 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 is the first time students will consider a fair game. You may want to explain that in a fair game, both players are expected to come out equally well in the long run.

Discussing and Debriefing the Activity Have a volunteer begin the discussion of Question 1. As needed, help students move beyond the fact that Tony wins 915 size 12{ { {9} over {"15"} } } {} of the time to focus on how much the players win. Students will likely use a “large number of trials” analysis to find that Tony loses, say, $30 over 150 games. As the conversation focuses on Tony’s expected value per turn, the idea of expected value being a negative number will probably occur to many students. Confirm that this is the standard way to express the idea that, over the long run, Tony loses money. Tony’s expected value per turn can be written as –$0.20 or –20¢. Through their calculations, students will see that Crystal’s expected value is the opposite of Tony’s. Once the two expected values have been determined, focus on the question of whether the game is fair. Ask, What does “fair” mean in the context of this game? Students will probably agree that a fair game is one in which the players come out even in the long run. This game is not fair, because Crystal comes out better in the long run. If the game were fair, what would Tony’s expected value be? Some students may be ready to answer this question. If not, you might then ask, How could you change the game to make it fair? Let volunteers offer suggestions. The smallest whole-number solution for the payoffs is for Tony to win $2 from Crystal when the dart lands on black and Crystal to win $3 from Tony when the dart lands on white. Any payoff in this 2:3 ratio will work. Have students now confirm that if the game were fair, Tony’s and Crystal’s expected values would both be $0.

Key Questions What does “fair” mean in the context of this game? If the game were fair, what would Tony’s expected value be? How could you change the game to make it fair?

Supplemental Activity Fair Spinners (reinforcement) asks students to modify the spinner from Spinner Give and Take to create another fair game.