Intent

In this exploratory activity, a spinner game is used as a probability model to introduce the notion of expected value. The approach is to consider a large number of trials in order to determine the average over the long run.

Mathematics

This activity draws upon students’ intuitive understanding of the law of large numbers as they consider what will happen in a game involving a back-and-forth payoff between two players that is determined by set probabilities. Students are asked to consider who will come out ahead over the long run. This experiential task begins their investigation into expected value, a description of the average points won (or lost) per turn over the long run. In this unit, the notion of over the long run will be emphasized to ground students’ reasoning about the concept and the computation of expected value.

Progression

Students begin by predicting the winner of a game and then carry out simulations to test their conjectures. Through this experience, students begin to develop a theoretical analysis of the game’s structure. Midway through the investigation, the class comes together to share ideas and to consider some of the quantitative conclusions that can be drawn.

Approximate Time

35 minutes

Classroom Organization

Pairs, followed by whole-class discussion

Materials

Paper clips

Doing the Activity

This activity is best completed in pairs. Demonstrate how to make a spinner, if necessary.

Discussing and Debriefing the Activity

Students will often neglect to consider that Al and Betty are paying each other in the game. This observation can be used to reengage students who are quick to decide they have finished the activity. Ask, If Betty gets her winnings from Al, what effect does that have on Al’s bank account (or wallet)?

Begin the discussion when all pairs have finished Questions 1 and 2.

For Question 1, ask several pairs to share their predictions and reasoning. How did you make your predictions for Question 1? Roughly speaking, students should articulate that although Betty wins more often, Al wins more money each time he wins.

Students should be expected to do some sort of quantitative work to support their predictions. They may do this in terms of a specific number of spins (for example, 100), estimating how many times each player would win and how much money each ends the game with.

For Question 2, ask several pairs for their results. Some pairs may have ended with Betty ahead, though most will have Al ahead. Ask if the fact that some pairs have Betty winning after 25 spins means their reasoning in Question 1 was incorrect. If necessary, call students’ attention to the phrase in the long run in Question 1. Students should gradually be developing and articulating the idea that although any result can occur in a small number of spins, in the long run the number of times each person wins will be roughly proportional to the probabilities.

Consider pooling all the pairs’ results; presumably Al will win about 1414 size 12{ { {1} over {4} } } {} of the time and Betty about 3434 size 12{ { {3} over {4} } } {} of the time. You might return to these numbers after the discussion of Question 3 to compute each player’s total winnings.

Answering Question 3 involves moving from probabilities to number of wins to amounts won to the difference in those amounts. Follow students’ lead in the discussion while you guide them toward an analysis that goes something like this.

1.What is Betty’s probability of winning? Al’s? From the shading of the spinner, Betty’s probability of winning a given spin is 3434 size 12{ { {3} over {4} } } {} and Al’s is 1414 size 12{ { {1} over {4} } } {}.

2.How many times out of 100 should Betty win? What about Al? If Betty and Al spin 100 times altogether, and the results follow the theoretical probabilities, we can expect Betty to win about 75 times and Al to win about 25 times.

Figure 1

3.How much do they each win? Betty would win $1 from Al in each of her 75 victories, and Al would win $4 from Betty in each of his 25 victories. So, in 100 spins, Betty will win about $75 from Al 75⋅$175⋅$1 size 12{ left ("75" cdot $1 right )} {} and Al will win about $100 from Betty 25⋅$425⋅$4 size 12{ left ("25" cdot $4 right )} {}.

4.How far ahead will Al be at the end? Over the course of 100 spins, Al will be about $25 ahead of where he began.

In the discussion, emphasize that in any particular set of spins, a variety of results are possible. You might ask, How many spins would Al have to win in order to come out ahead? Students can use guess-and-check to see that he needs to win at least 21 out of 100 spins to come out ahead of Betty.

The fact that Al wins even if the results are slightly off from the probabilities is part of the strength of the “large number of spins” approach. Students should gradually develop an intuitive sense that the greater the number of spins, the more likely it is that Al will come out ahead.

Have students conduct the same analysis with at least two different numbers of spins. They should see in each case that Al comes out ahead and that he wins more money if they play more spins. In fact, for twice as many spins, he wins twice as much.

You can ask what remains the same in each case, besides the fact that Al is ahead. You might ask them to consider what it may mean that, for example, for twice as many spins, this analysis shows Al winning twice as much. This question addresses the idea directly: How much did Al win per spin, on the average? Students should see that in each case, the average is 25¢. In other words, his end result is the same as if he had won 25¢ on each spin.

In the upcoming activity Mia’s Cards, the term expected value is introduced as a shorthand for the idea of average per spin (in the long run). For now, the goal is that students begin to agree that average per spin is a good way to measure what happens in the long run, as the outcome is the same no matter how many spins are made.

Ask, How does this problem relate to the game of Pig? You may want to use this question as a topic for focused free writing and then have students share their ideas. As needed, remind students that in the game of Pig, they are also interested in what happens in the long run. Specifically, they want to know which strategy is likely to produce the most points per turn in the long run. You can mention that finding the probabilities for a Pig strategy is much harder than finding the probabilities for this simple spinner game.

Key Questions

If Betty gets her winnings from Al, what effect does that have on Al’s bank account (or wallet)?

How did you make your predictions for Question 1?

What results did you get for Question 2?

Does the fact that some pairs showed Betty winning mean that the reasoning was incorrect?

What is Betty’s probability of winning? Al’s?

How many times out of 100 should Betty win? What about Al?

How much do Betty and Al each win?

How far ahead is Al at the end?

How many spins would Al have to win in order to come out ahead?

What happens if you use a different total number of spins?

How much did Al win per spin, on the average?

How does this situation relate to the game of Pig?