Intent

Students will employ their available tools to compute an expected value and compare it to a simulated result.

Mathematics

In the typical payment plan, a customer pays the same amount each week or month for a given service. In this activity, a newspaper carrier is paid by a particular customer by choosing, each week, two bills from a bag containing five $1 bills and one $10 bill. The task is to determine the average weekly payment—that is, the expected value—and compare it to the results of a simulation.

Progression

The activities in In the Long Run bring together all of the ideas developed thus far and set students up for the activities in Analyzing a Game of Chance, in which they will return to the unit problem.

Approximate Time

5 minutes for introduction

25 minutes for activity (at home or in class)

30 minutes for discussion

Classroom Organization

Individuals, followed by whole-class discussion

Doing the Activity

You may want to brainstorm with students how they could carry out this simulation if they didn’t have dollar bills—for example, by using five red cubes and one blue cube.

Discussing and Debriefing the Activity

Have students share their simulation methods and then compile the results. If the average of the data is close to the theoretical expected value of $5 per week, the class will probably not be able to draw any clear conclusions about which payment method produces more money for the newspaper carrier.

The class may have used a variety of approaches to find the theoretical probabilities of getting $2 and of getting $11. However students find the probabilities, the “large number of trials” method can then be used to determine the expected value. As students share how they found the probabilities, help them to focus on the distinction between possible outcomes and equally likely outcomes.

Labeling each individual bill can highlight the fact that the probability of drawing a $1 bill is not the same as the probability of drawing the $10 bill, but that the probability of drawing a particular $1 bill is the same as the probability of drawing the $10 bill. In other words, the equally likely outcomes for a given draw are the individual bills, not the amounts.

In the area models below, T represents the $10 bill, and O1O1 size 12{O rSub { size 8{1} } } {} , O2O2 size 12{O rSub { size 8{2} } } {}, O3O3 size 12{O rSub { size 8{3} } } {}, O4O4 size 12{O rSub { size 8{4} } } {}, and O5O5 size 12{O rSub { size 8{5} } } {} represent the five $1 bills. If you use subscript notation, you might mention how to read subscripted variables (for example, “oh sub one”). To create an area model, it may help to think of the bills as drawn one after the other, rather than both at once. After the first draw, the diagram could look like this.

Figure 1

For each choice of the first bill, there are five possibilities for the second bill, so each column is divided into five sections, yielding 30 equal parts that represent the equally likely outcomes. The ten shaded boxes represent the cases in which a $10 bill and a $1 bill have been drawn, for a total of $11.

Figure 2

Thus, P$11=1030=13P$11=1030=13 size 12{P left ($"11" right )= { {"10"} over {"30"} } = { {1} over {3} } } {} and P$2=2030=23P$2=2030=23 size 12{P left ($2 right )= { {"20"} over {"30"} } = { {2} over {3} } } {}.

The diagrams below avoid referring to individual bills.

Figure 3

Figure 4

Some students may have constructed a tree diagram to represent the situation. If not, you might suggest this to reinforce the tool, which was introduced earlier. Such a tree diagram might look like this.

Figure 5

It may be less cumbersome to make a separate diagram for each set of subbranches.

Figure 6

Altogether, there are 30 possible paths, which correspond to the 30 boxes in the area model. Because the paths are all equally likely, each path has a probability of 1313 size 12{ { {1} over {3} } } {} . Students can count paths to find the probability of each payment outcome.

Ask whether anyone used another approach. Some students may have listed the possible combinations, without regard to which bill was picked first. Such a list might look like this.

TO 1 , TO 2 , TO 3 , TO 4 , TO 5 TO 1 , TO 2 , TO 3 , TO 4 , TO 5 size 12{ ital "TO" rSub { size 8{1} } , ital "TO" rSub { size 8{2} } , ital "TO" rSub { size 8{3} } , ital "TO" rSub { size 8{4} } , ital "TO" rSub { size 8{5} } } {}

O 1 , O 2 , O 1 , O 3 , O 1 , O 4 , O 1 , O 5 O 1 , O 2 , O 1 , O 3 , O 1 , O 4 , O 1 , O 5 size 12{O rSub { size 8{1} } ,O rSub { size 8{2} } ,O rSub { size 8{1} } ,O rSub { size 8{3} } ,O rSub { size 8{1} } ,O rSub { size 8{4} } ,O rSub { size 8{1} } ,O rSub { size 8{5} } } {}

O 2 , O 3 , O 2 , O 4 , O 2 , O 5 O 2 , O 3 , O 2 , O 4 , O 2 , O 5 size 12{O rSub { size 8{2} } ,O rSub { size 8{3} } ,O rSub { size 8{2} } ,O rSub { size 8{4} } ,O rSub { size 8{2} } ,O rSub { size 8{5} } } {}

O 3 , O 4 , O 3 , O 5 O 3 , O 4 , O 3 , O 5 size 12{O rSub { size 8{3} } ,O rSub { size 8{4} } ,O rSub { size 8{3} } ,O rSub { size 8{5} } } {}

O 4 , O 5 O 4 , O 5 size 12{O rSub { size 8{4} } ,O rSub { size 8{5} } } {}

As these 15 possibilities are equally likely, students can count to find the probabilities. You may want to ask why this method produces 15 possibilities, while the other methods show 30. The reason is that the list of combinations ignores different orders. For example, it shows TO1TO1 size 12{TO rSub { size 8{1} } } {} but does not also show O1TO1T size 12{O rSub { size 8{1} } T} {}.

However students determine the probabilities, they can use the “large number of trials” method to find the expected value. For example, for 300 trials, the carrier will get $11 about 100 times ( 1313 size 12{ { {1} over {3} } } {} of the cases) and $2 about 200 times ( 2323 size 12{ { {2} over {3} } } {} of the cases). Thus the carrier would expect to get $1,500 for 300 weeks, an expected value of exactly $5 a week. In other words, it turns out that the two plans have the same expected value.

Ask students, Which payment plan would you choose, and why? Although the expected value is the same for both plans, money may not be the deciding factor for some students as to which method is better. Some students may argue that the sense of adventure that accompanies the second plan makes it preferable even if it were to produce less money in the long run, while others may prefer the security of the first plan even if it were to earn less.