In this activity, students conduct their first simulation, collecting, displaying, and summarizing data to test a conjecture about probability. The activity draws upon their intuitive sense of probability and engages them in thinking more precisely about the topic. This is one of several activities in which intuitive notions about probability can be at odds with experimental or theoretical analyses.

When rolling two dice, what is the average number of rolls needed to roll a double? In this activity, students approach this question through collecting and analyzing data. By rolling two dice repeatedly, they collect many examples of how many rolls it can take to get a double.

Up to now, students have been exploring the unit problem informally. Now they begin a series of explorations designed to help them develop the tools they will need to answer the unit problem. The in-class portion of this activity has several parts: reviewing the homework, gathering class data, and graphing the class data.

5 minutes for introduction

20 minutes for activity (at home or in class)

25 minutes to collect and analyze class data

Individuals, then small groups to review data, test Equation Editor: z2+33=5.5z2+33=5.5 size 12{z rSup { size 8{2} } +3 rSup { size 8{3} } = sqrt {5 "." 5} } {} + 4. Then the whole class gathers to analyze data. This is a test of Equation Editor: z2+33=5.5z2+33=5.5 size 12{z rSup { size 8{2} } +3 rSup { size 8{3} } = sqrt {5 "." 5} } {} + 4.

x 2 + y 2 = z 1 + z 2 x 2 + y 2 = z 1 + z 2 size 12{x rSup { size 8{2} } +y rSup { size 8{2} } =z rSub { size 8{1} } +z rSub { size 8{2} } } {}

1 2 2 ³ AC ¯ BC → ˆ ∑ i = 0 n 1 2 2 ³ AC ¯ BC → ˆ ∑ i = 0 n alignl { stack { size 12{ { {1} over {2} } sqrt {2} ³ {overline { ital "AC"}} widehat widevec { ital "BC"} } {} # size 12{ Sum cSub { size 8{i=0} } {n} } {} } } {}

For a discussion of simulating class data, click here. For a Fathom file that can be used as a demonstration of the process followed during this activity, click here.

Before assigning the activity:

How would you do this experiment?

Why is it important to get genuine data when doing activities like this?

How do you compute the average of a set of numbers?

During the class discussion:

How did you make your prediction?

What were the highest and lowest number of rolls you got?

How might you graph one student’s outcomes?

Can you visualize the average in terms of the graph?

How might we graph the outcomes of the entire class?

Is 1 a likely outcome?