1
Schoolopoly
Mathematical Goals: Teachers will be able to
 Understand how to simulate randomness with pseudo-random number generators, and
use data from repeated trials to make informed estimates of reasonable intervals.
 Consider the effect of sample size and variability when comparing empirical results with
expected theoretical outcomes.
Pedagogical Goals: Teachers will be able to
 Consider the benefits and drawbacks for using computing tools to generate pseudorandom numbers.
 Examine the benefits and drawbacks of using multiple representations to examine data
resulting from a random process.
 Compare the use of graphing calculators and spreadsheets for conducting simulations and
analyzing data generated from random processes.
Technological Goals: Teachers will be able to use a technological tool to
 Learn the importance of seed values in pseudo-random number generators.
 Use graphing calculators to generate a sequence of random integers, save the sequence as
a List, and construct a histogram of the results.
 Use spreadsheets to generate a sequence of random integers, tally the occurrence of
specific values and construct graphs of the results.
Mathematical Practices:
 Make sense of problems and persevere in solving them.
 Reason abstractly and quantitatively.
 Construct viable arguments and critique the reasoning of others.
 Model with mathematics.
 Use appropriate tools strategically.
 Attend to precision.
 Look for and make use of structure.
Length of session: 90 minutes
Materials needed: Computer with TinkerPlots, CalibratedCubes.tp file, DazzlingDice.tp file,
DeltasDice.tp file, DiceDepot.tp file, DiceRUs.tp file, HighRollersInc.tp file, Graphing
calculators, Schoolopoly Participant handout
Overview:
In this session, participants will use TinkerPlots simulations to determine which dice company actually sells fair die.
Then, participants will consider how simulations may be represented with real objects (such as coins) and how we
can use a graphing calculator to model a real world situation to collect a large number of trials rather than having to
flip a coin repeatedly. Participants will also consider the “randomness” of technology tools.
Estimated #
Activity
Adapted from:
Lee, H. S., Hollebrands, K. F., & Wilson, P. H. (2010). Designing and using probability simulations. In Preparing to teach mathematics with
technology: An integrated approach to data analysis and probability (103-125). Dubuque: Kendall Hunt.
Tarr, J. E., Stohl Lee, H., & Rider, R. (2006). When data and chance collide: Drawing inferences from simulation data. In G. F. Burrill & P. C.
Elliott (Eds.), Thinking and reasoning with data and chance: Sixty-eighth NCTM yearbook (pp. 139-150). Reston, VA: National
Council of Teachers of Mathematics.
2
of Minutes
5 minutes
25 minutes
Introduction
 Probabilistic reasoning is difficult and our intuitions often lead us astray.
Probability is based on randomness and thus removes cause and effect
relationships we often want to look for.
 Engaging in simulation activities that are grounded in real world contexts can
give students a better intuition for probability than they typically develop
when computing a probability using rules and formulas.
Schoolopoly
Consider the following problem:
Suppose your school is planning to create a board game modeled on the
classic game of Monopoly. The game is to be called Schoolopoly and, like
Monopoly, will be played with dice. Because many copies of the game
expect to be sold, companies are competing for the contract to supply dice
for Schoolopoly. Some companies have been accused of making poor quality
dice and these are to be avoided since players must believe the dice they are
using are actually “fair.” Each company below has provided dice for
analysis.
 Calibrated Cubes
 Dazzling Dice
 Delta’s Dice
 Dice Depot
 Dice R’ Us
 High Rollers, Inc.
Working with a partner, investigate whether the dice sent from each company is
fair or biased.
Participants should find only one company with fair dice. They should be able to
support their reasoning with data (graphs, tables, etc.). Allow participants to share
their findings with each other and discuss how this task might help motivate
students to want to generate their own simulations for determining the probability
of events.
Questions to consider:
1. Do you believe the dice from each company are fair or biased?
Which company would you recommend purchasing dice from? The
only fair die are from Dice R’ Us. The other companies all have
biased die and should not be recommended as the provider of dice
for the game.
2. What compelling evidence do you have that the dice you of each
company are fair or unfair? Results may include tables of values,
histograms, pie charts, etc.
3. Use your data to estimate the probability of each outcome, 1-6, of
Adapted from:
Lee, H. S., Hollebrands, K. F., & Wilson, P. H. (2010). Designing and using probability simulations. In Preparing to teach mathematics with
technology: An integrated approach to data analysis and probability (103-125). Dubuque: Kendall Hunt.
Tarr, J. E., Stohl Lee, H., & Rider, R. (2006). When data and chance collide: Drawing inferences from simulation data. In G. F. Burrill & P. C.
Elliott (Eds.), Thinking and reasoning with data and chance: Sixty-eighth NCTM yearbook (pp. 139-150). Reston, VA: National
Council of Teachers of Mathematics.
3
the dice for each company. See chart below. Participants’ values
may differ slightly but if enough trials were conducted, their
empirical probabilities may be close to the theoretical probability.
Company
Calibrated
Cubes
Dazzling
Dice
Delta’s
Dice
Dice
Depot
Dice R’
Us
High
Rollers,
Inc.
10 minutes
Weight/Theoretical Probability
3
3
3
0.15
0.15
0.15
2
3
3
0.125
0.1875
0.1875
2
3
2
0.1333
0.2
0.1333
2
3
4
0.1111
0.1667
0.2222
1
1
1
0.1667
0.1667
0.1667
4
5
5
0.16
0.2
0.2
3
0.15
3
0.1875
3
0.2
4
0.2222
1
0.1667
5
0.2
3
0.15
3
0.1875
2
0.1333
3
0.1667
1
0.1667
1
0.04
5
0.25
2
0.125
3
0.2
2
0.1111
1
0.1667
5
0.2
Note: The weights and corresponding theoretical probabilities are shown in the
table below. To alter the task, you may unlock each of the samplers by typing
“Fair” as the password and then selecting to show the contents of the sampler.
Simulating Randomness in Technology Tools
 Technology tools use deterministic algorithms to generate “random”
numbers. They all start with an initial input, or seed value, which tells the
computer where in the list of numbers to begin its computation. Thus, if
you know the seed value and the algorithm being used, it is possible to
predict the output. Therefore, we call these pseudo-random number
generators since the process is not truly random. However, if you do not
know the seed value or algorithm used, it is unlikely you can predict the
output of a computer’s “random” function so pseudo-random generators
are accepted for simulations where we want to model a random process.
 Let’s consider a graphing calculator. When you first take the graphing
calculator out of the package, the seed value is set to 0. So, prior to using
the Rand function, you will want to change the seed value on each
calculator to ensure students receive different outputs.
 To set the seed value:
1. Type in a value – One way to choose a seed value is for each student
to type in his or her birthday as a 4-digit number followed by a
number assigned to them by counting off (e.g., 120714 would be a
student whose birthday is December 7th and is the 14th student in the
class).
2. Press the “STO” key (store key).
3. Press the “MATH” key.
Adapted from:
Lee, H. S., Hollebrands, K. F., & Wilson, P. H. (2010). Designing and using probability simulations. In Preparing to teach mathematics with
technology: An integrated approach to data analysis and probability (103-125). Dubuque: Kendall Hunt.
Tarr, J. E., Stohl Lee, H., & Rider, R. (2006). When data and chance collide: Drawing inferences from simulation data. In G. F. Burrill & P. C.
Elliott (Eds.), Thinking and reasoning with data and chance: Sixty-eighth NCTM yearbook (pp. 139-150). Reston, VA: National
Council of Teachers of Mathematics.
4
4. Choose “PRB.”
5. Select 1:rand.
6. Press enter and the seed value will be displayed.

20 minutes
Let’s simulate a die toss on the graphing calculator. The command randInt
will allow us to choose an integer between two other integers for a given
number of times. For example randInt(1,6,5) will choose a number between
1 and 6 five times. The outcome is based on an equiprobable die.
 Questions to consider:
1. Would you prefer to discuss the importance of a seed value with
students or to set all of the graphing calculators with different seed
values yourself and not discuss the issues with students? Explain
your choice. Having students understand how to seed might make
learning simulation procedures easier for students, as they will be
able to follow along with the teacher or confirm with a neighboring
student. Some teachers may prefer to set all the GCs with a different
seed value in order to avoid class discussions with students who may
doubt that computers can really produce random numbers.
However, teachers that make this choice are knowingly avoiding the
fact that computers must use an algorithm to produce the sequence
of numbers.
2. How could you sue the fact that many calculators and computers
generate the same list of pseudo-random numbers given the same
initial seed value to generate discussions with students about
randomness in general and the use of computers to simulate
probability experiments? Having an algorithm to produce pseudorandom numbers is contrary to the very idea of randomness. This
contrast might help students understand stochastic versus
deterministic. If every graphing calculator produces the same
output, that can demonstrate a deterministic algorithm. Inputting the
different seed values can then illustrate how a stochastic process can
be imitated, although it is not truly random. Because random
number generators can be seeded differently, they can model
random phenomenon using computers. Users of computers need to
be aware of the limitations of the random number generators used
and the importance of seed values, particularly if one is creating a
simulation tool (i.e., programming their own application). However,
the algorithms used by most computing tools are so sophisticated
that they can often simulate a random process more “reliably” than
if students were tossing a coin by hand.
Using Data to Estimate Probabilities and Design Simulations
 Consider the graph below.
Adapted from:
Lee, H. S., Hollebrands, K. F., & Wilson, P. H. (2010). Designing and using probability simulations. In Preparing to teach mathematics with
technology: An integrated approach to data analysis and probability (103-125). Dubuque: Kendall Hunt.
Tarr, J. E., Stohl Lee, H., & Rider, R. (2006). When data and chance collide: Drawing inferences from simulation data. In G. F. Burrill & P. C.
Elliott (Eds.), Thinking and reasoning with data and chance: Sixty-eighth NCTM yearbook (pp. 139-150). Reston, VA: National
Council of Teachers of Mathematics.
5

The graph has three schools highlighted. The rates for each school describe
the actual percent of freshman from an incoming class of full-time students
(population) that enrolled in courses for a second year. College
administrators could simply use the past retention rates to calculate a
predicted number of freshman who will return the next year. However,
since we can expect some variability from year to year with different size
classes, it is helpful to use retention rates as an estimate of the probability
for any given freshman returning to school their sophomore year. This can
help administrators to make more informed estimates of a reasonable
number of returning sophomores.

Questions to consider:
1. Use the retention rates at Chowan College, NC Central and NC State
University to estimate the probability that a randomly chosen
freshman will continue into their second year at each school.
Estimates might be Chowan—0.5; NCCU—0.8; NCSU—0.9
2. Describe how you would use a coin to simulate the experiment of
deciding whether or not any given freshman will continue on the
next year at Chowan College. What other objects could you use to
conduct a simulation? Since the estimated probability of graduation
at Chowan is 0.5, flipping a fair coin can model that scenario if
heads is taken to be “Student returns” and tails is taken to be
“Student does not return.”
3. If you conduct the simulation you described above with 30 trials to
represent the number of freshman, what is a reasonable amount of
these 30 freshman you could expect to return the following year?
Why? Most students will predict around 15, because 0.5*30=15
successes, though one should expect some variability and an interval
of about 10-20 is reasonable.
4. If you repeat a simulation of 30 trials several times (decide how
many times), what similarities or differences do you expect across
the results from the different samples of 30 trials? The results of
each sample may vary considerably from 15. Each of the samples
will likely not be 15, and may have a large spread due to the small
sample size. Note: Students may NOT actually anticipate a large
spread, this is ok. They may predict little variation from 15 such as
Adapted from:
Lee, H. S., Hollebrands, K. F., & Wilson, P. H. (2010). Designing and using probability simulations. In Preparing to teach mathematics with
technology: An integrated approach to data analysis and probability (103-125). Dubuque: Kendall Hunt.
Tarr, J. E., Stohl Lee, H., & Rider, R. (2006). When data and chance collide: Drawing inferences from simulation data. In G. F. Burrill & P. C.
Elliott (Eds.), Thinking and reasoning with data and chance: Sixty-eighth NCTM yearbook (pp. 139-150). Reston, VA: National
Council of Teachers of Mathematics.
6
30 minutes
13, 14, 15, 16, 17, 18 freshmen returning to school in each sample of
30 trials.
5. Use a coin to conduct a simulation of several samples of 30 trials
and record your results. How do the results compare with what you
anticipated? Students might be surprised to find more variability in
the proportion of heads (or tails) in the distribution of sample
proportions from samples of 30 trials, than in what they anticipated.
For example one such list of frequency of freshman returning and
corresponding proportions could include (actual results from 10
samples of 30 trials)
{13, 16, 16, 10, 10, 13, 18, 17, 13, 11}
{0.4333, 0.5333, 0.5333, 0.3333, 0.3333, 0.4333, 0.6, 0.5666, 0.4333,
0.3666}
6. What are some of the potential benefits and drawbacks of using a
real world context like the freshman retention rate for introducing
probability to students, especially terminology such as outcomes,
sample space, experiment, trial, event, sample, and sample size? A
real world context can help students see how probability is applied
in a real world setting and can sometimes help students be engaged
in the problem. The real world context can also help student attach
experiential meaning to terms such as outcome or sample space. The
real world context can help students think about what the possible
different outcomes are that make up the sample space. This also
brings up important issues of using a probability to model a real
world context in that assumptions have to be made about the
repeatability of an experiment under identical constraints. It is often
difficult to distinguish between trial and sample, and a real context
can help with that. With the freshman retention context, every
freshman represents a trial with a 50% chance of returning and the
size of freshman class is a sample of n freshman (trials). Mapping
the real context onto the coin toss simulation can help students
simplify the experiment. However, this can also be a lot to think
about in the problem and some students may still confuse some of
the terminology.
Simulating Events with a Graphing Calculator
 Still considering the retention rate situation of Chowan College, recall our
sample space has two possible outcomes: student returns and student does
not return. To simulate this situation using our graphing calculators, we
will use the randInt function assigning two consecutive integers to
represent the two outcomes.

Questions to consider:
1. Suppose the freshman class at Chowan College has 500 students.
What command would you use on the graphing calculator to conduct
a simulation of whether or not each of the 500 freshmen stays in
Adapted from:
Lee, H. S., Hollebrands, K. F., & Wilson, P. H. (2010). Designing and using probability simulations. In Preparing to teach mathematics with
technology: An integrated approach to data analysis and probability (103-125). Dubuque: Kendall Hunt.
Tarr, J. E., Stohl Lee, H., & Rider, R. (2006). When data and chance collide: Drawing inferences from simulation data. In G. F. Burrill & P. C.
Elliott (Eds.), Thinking and reasoning with data and chance: Sixty-eighth NCTM yearbook (pp. 139-150). Reston, VA: National
Council of Teachers of Mathematics.
7
school with a 50% estimate for the probability for retention? Explain
each part of the command in terms of the context of the situation.
One possible command is randInt(0, 1, 500). This function generates
500 random integers of value 0 or 1. In this simulation, 0 could
represent “student does not return”, 1 could represent a “student
returns”, and 500 is the size of the freshman class.
2. Given a 50% estimate for the probability for retention, out of 500
freshmen, what is a reasonable interval for the proportion of
freshmen you would expect to return the following year? Defend
your expectation. Estimates will vary. After running 10 simulations,
one could expect between 46% and 54% (about 228 and 276
freshman). Some may guess something as small as 49%-51% or a
larger more unreasonable (too wide considering 500 trials) spread
would be 40-60%.


Now we will model this situation using our graphing calculators. We will
store the set of numbers as a list so that we can also view a histogram of
our output and observe what happens when we repeat the simulation.
o Type randInt(0, 1, 500)  L1. Here we will generate a list of 500
random integers between 0 and 1. A result of 0 will represent a
student does not return while a result of 1 represents the case when a
student returns.
o When you press enter, you will see the list of the 500 random
integers. Now we want to see a histogram of this to see the
frequency of 0s and 1s.
o Activate Plot1 on the StatPlot menu.
o Select the Histogram icon for the Type of plot and make sure the
Xlist displays L1.
o Set the window to have range from 0 to 2 for x (scale of 1) and -50
to 350 for y (scale of 50) with x resolution 1.
o Now if you press the graph key, you should see the histogram.
Questions to consider:
3. Explain why the values used in the Window setting create the
appropriate graphical display of two bars with the frequency of the
0s in the first and the frequency of the 1s in the second. In your
response, explain why a y max of 350 is appropriate. The first
histogram bin represents all observations below 1. The second bin
contains all observations from 1 and up to 2, but not including 2.
Creating a domain from 0 to 2 allows for both of those bins to be
visible. The Ymax of 350 is a reasonable estimate of the maximum
frequency expected for either outcome and ensures that the window
will be high enough to capture the height of each of the bins.
4. Determine the proportion of freshmen who will return to Chowan
College next year. Answers will vary but should be near 0.50.
5. For our problem we are interested in how much the proportion of
Adapted from:
Lee, H. S., Hollebrands, K. F., & Wilson, P. H. (2010). Designing and using probability simulations. In Preparing to teach mathematics with
technology: An integrated approach to data analysis and probability (103-125). Dubuque: Kendall Hunt.
Tarr, J. E., Stohl Lee, H., & Rider, R. (2006). When data and chance collide: Drawing inferences from simulation data. In G. F. Burrill & P. C.
Elliott (Eds.), Thinking and reasoning with data and chance: Sixty-eighth NCTM yearbook (pp. 139-150). Reston, VA: National
Council of Teachers of Mathematics.
8
freshmen returning to Chowan College will vary from the expected
50%. Use the skills you have learned to repeat the simulation of
deciding the retention of 500 freshman many times. Record the
proportion of freshmen that continue on next year for each of the
samples of 500 trials. How many of the sample proportions fall
within the interval you predicted in question 2? Discuss why the
results may or may not have been consistent with your earlier
prediction. Some teachers may be surprised that the range from the
simulation data does not match their initial prediction. They may
have predicted too wide or too narrow of an interval.
6. Recall that in our simulation, each freshman had a 50% chance of
returning to school. Did you get exactly 50% of the freshmen
returning in the simulation? Discuss how and why the empirical
proportions from the simulation may have varied from 50%. Not
often because of expected variation. Note: it is not expected at this
point for teachers to actually calculate probabilities. However, the
probability of obtaining exactly 50% (250 freshman, P (250)
=.0357) is smaller than the probability of obtaining values close to
250 (e.g., (P (249 or 251) =.0355 + .0355 = .071). However, the
empirical results clumped around the value of 0.5.
7. If we reduced the number of trials to 200 freshmen, what do you
anticipate would happen to the interval of proportions from the
empirical data around the theoretical probability of 50%? Why?
Conduct a few samples with 200 trials and compare your results
with what you anticipated. Because the sample size decreased, the
sample proportions will vary more and the interval will be wider.
8. If we increased the number of trials to 999 freshmen, what do you
anticipate would happen to the interval of proportions from the
empirical data around the theoretical probability of 50%? Why?
Conduct a few samples with 999 trials and compare your results
with what you anticipated. Because the sample size increased, the
sample proportions will vary less and the interval will be narrower.
9. Based on your experience with the simulations and the empirical
data you collected, what would be a reasonable interval for the
proportion of freshmen that administrators can expect to return to
Chowan College for freshmen class sizes of 200, 500, and 999?
Explain your predictions. For ten trials of each, a sample response:
From 0.4 to 0.535 with 200 freshmen
From 0.456 and .522 with 500 freshmen
From 0.476 and 0.526 with 999 freshmen.
As the number of freshmen in a trial increases, more is known about
population of all freshmen that attend Chowan. Thus, we can expect
less variability in the proportion of returning students.
10. Discuss why it might be beneficial to have students simulate the
freshman retention problem for several samples of sample size 500,
Adapted from:
Lee, H. S., Hollebrands, K. F., & Wilson, P. H. (2010). Designing and using probability simulations. In Preparing to teach mathematics with
technology: An integrated approach to data analysis and probability (103-125). Dubuque: Kendall Hunt.
Tarr, J. E., Stohl Lee, H., & Rider, R. (2006). When data and chance collide: Drawing inferences from simulation data. In G. F. Burrill & P. C.
Elliott (Eds.), Thinking and reasoning with data and chance: Sixty-eighth NCTM yearbook (pp. 139-150). Reston, VA: National
Council of Teachers of Mathematics.
9
as well as sample sizes of 200 and 999. The first simulation might
not be very far from the theoretical probability; several simulations
increased the chances of students seeing a more extreme case. Also,
conducting the simulation several times might help them understand
sampling variability.
11. An important aspect of conducting a simulation of a repeated
random event is to be sure that students have a conceptual
understanding of how the simulation and the commands used in the
computing tool represents the context of the problem. Consider the
following context:
Suppose a university gives away token spirit gifts to all
incoming freshmen. As they check in to pick up their class
schedule, they get to randomly choose three cards. Each card
displays a different gift: keychain, window decal for a car, tshirt with college logo. Most freshmen would prefer the t-shirt.
Explain how you would help students use the graphing calculator to
simulate this context. Explicitly describe what the commands
represent and how the students should interpret the results. Let 0
represent the T-shirt, 1 represent the window decal, and 2 represent
L1 would simulate
randomly selecting 100 cards. The numbers recorded in L1 would
represent the prizes that would be awarded for each card. Counting
the number of 0’s would correspond to the number of T-shirts
awarded.
12. Suppose that after conducting the above simulations, a student gets a
result that a t-shirt was chosen 20% of the time. The student is
surprised and claims that this proportion is too low from the
expected 33.3%. What might this student be misunderstanding?
What are some questions you could pose and further simulations you
could suggest that might help this student? The student might not
understand sampling variability, especially with a sample size of 30
and three possible outcomes. They may also not understand that
although 20% is 13.33% less than the expected 33.33%, it is only a
result of 4 less than the expected 10 students choosing a t-shirt.
Sometimes it is helpful to ask students to consider sample sizes that
are at a lower extreme (like 4 or 10) to get students to imagine and
experience the variation that can occur from what is expected. One
possible question might be “If you flipped a fair coin 4 times, how
many heads would you expect? What about 10 times? Could you
expect 2 out of 4 or 5 out of 10 every time you tried this?” Students
can do these experiments several times to see the variation that
occurs. Having them then do repeated samples of 30 freshman
choosing a card can help them notice that an outcome of 6 for any of
the 3 possible choices can occur quite easily, especially if you have
a whole class of students repeating the simulation several time until
Adapted from:
Lee, H. S., Hollebrands, K. F., & Wilson, P. H. (2010). Designing and using probability simulations. In Preparing to teach mathematics with
technology: An integrated approach to data analysis and probability (103-125). Dubuque: Kendall Hunt.
Tarr, J. E., Stohl Lee, H., & Rider, R. (2006). When data and chance collide: Drawing inferences from simulation data. In G. F. Burrill & P. C.
Elliott (Eds.), Thinking and reasoning with data and chance: Sixty-eighth NCTM yearbook (pp. 139-150). Reston, VA: National
Council of Teachers of Mathematics.
10
someone get s a result of 6/30 for one of the three choices.
Adapted from:
Lee, H. S., Hollebrands, K. F., & Wilson, P. H. (2010). Designing and using probability simulations. In Preparing to teach mathematics with
technology: An integrated approach to data analysis and probability (103-125). Dubuque: Kendall Hunt.
Tarr, J. E., Stohl Lee, H., & Rider, R. (2006). When data and chance collide: Drawing inferences from simulation data. In G. F. Burrill & P. C.
Elliott (Eds.), Thinking and reasoning with data and chance: Sixty-eighth NCTM yearbook (pp. 139-150). Reston, VA: National
Council of Teachers of Mathematics.
11
Schoolopoly
Participant Handout
Consider the following problem:
Suppose your school is planning to create a board game modeled on the classic game of
Monopoly. The game is to be called Schoolopoly and, like Monopoly, will be played with dice.
Because many copies of the game expect to be sold, companies are competing for the contract to
supply dice for Schoolopoly. Some companies have been accused of making poor quality dice
and these are to be avoided since players must believe the dice they are using are actually “fair.”
Each company below has provided dice for analysis.
 Calibrated Cubes
 Dazzling Dice
 Delta’s Dice
 Dice Depot
 Dice R’ Us
 High Rollers, Inc.
Working with a partner, investigate whether the dice sent from each company is fair or biased.
Questions to consider:
1. Do you believe the dice from each company are fair or biased? Which company
would you recommend purchasing dice from?
2. What compelling evidence do you have that the dice you of each company are fair or
unfair?
3. Use your data to estimate the probability of each outcome, 1-6, of the dice for each
company.
Simulating Randomness in Technology Tools
Questions to consider:
1. Would you prefer to discuss the importance of a seed value with students or to set all
of the graphing calculators with different seed values yourself and not discuss the
issues with students? Explain your choice.
2. How could you sue the fact that many calculators and computers generate the same
list of pseudo-random numbers given the same initial seed value to generate
discussions with students about randomness in general and the use of computers to
simulate probability experiments?
Adapted from:
Lee, H. S., Hollebrands, K. F., & Wilson, P. H. (2010). Designing and using probability simulations. In Preparing to teach mathematics with
technology: An integrated approach to data analysis and probability (103-125). Dubuque: Kendall Hunt.
Tarr, J. E., Stohl Lee, H., & Rider, R. (2006). When data and chance collide: Drawing inferences from simulation data. In G. F. Burrill & P. C.
Elliott (Eds.), Thinking and reasoning with data and chance: Sixty-eighth NCTM yearbook (pp. 139-150). Reston, VA: National
Council of Teachers of Mathematics.
12
Using Data to Estimate Probabilities and Design Simulations
Consider the graph below.
The graph has three schools highlighted. The rates for each school describe the actual percent of
freshman from an incoming class of full-time students (population) that enrolled in courses for a
second year. College administrators could simply use the past retention rates to calculate a
predicted number of freshman who will return the next year. However, since we can expect some
variability from year to year with different size classes, it is helpful to use retention rates as an
estimate of the probability for any given freshman returning to school their sophomore year. This
can help administrators to make more informed estimates of a reasonable number of returning
sophomores.
Questions to consider:
1. Use the retention rates at Chowan College, NC Central and NC State University to
estimate the probability that a randomly chosen freshman will continue into their
second year at each school.
2. Describe how you would use a coin to simulate the experiment of deciding whether or
not any given freshman will continue on the next year at Chowan College. What other
objects could you use to conduct a simulation?
3. If you conduct the simulation you described above with 30 trials to represent the
number of freshman, what is a reasonable amount of these 30 freshman you could
expect to return the following year? Why?
4. If you repeat a simulation of 30 trials several times (decide how many times), what
similarities or differences do you expect across the results from the different samples
of 30 trials?
5. Use a coin to conduct a simulation of several samples of 30 trials and record your
results. How do the results compare with what you anticipated?
6. What are some of the potential benefits and drawbacks of using a real world context
like the freshman retention rate for introducing probability to students, especially
terminology such as outcomes, sample space, experiment, trial, event, sample, and
sample size?
Adapted from:
Lee, H. S., Hollebrands, K. F., & Wilson, P. H. (2010). Designing and using probability simulations. In Preparing to teach mathematics with
technology: An integrated approach to data analysis and probability (103-125). Dubuque: Kendall Hunt.
Tarr, J. E., Stohl Lee, H., & Rider, R. (2006). When data and chance collide: Drawing inferences from simulation data. In G. F. Burrill & P. C.
Elliott (Eds.), Thinking and reasoning with data and chance: Sixty-eighth NCTM yearbook (pp. 139-150). Reston, VA: National
Council of Teachers of Mathematics.
13
Simulating Events with a Graphing Calculator
Still considering the retention rate situation of Chowan College, recall our sample space has two
possible outcomes: student returns and student does not return. To simulate this situation using
our graphing calculators, we will use the randInt function assigning two consecutive integers to
represent the two outcomes.
Questions to consider:
1. Suppose the freshman class at Chowan College has 500 students. What command
would you use on the graphing calculator to conduct a simulation of whether or not
each of the 500 freshmen stays in school with a 50% estimate for the probability for
retention? Explain each part of the command in terms of the context of the situation.
2. Given a 50% estimate for the probability for retention, out of 500 freshmen, what is a
reasonable interval for the proportion of freshmen you would expect to return the
following year? Defend your expectation.
Model using your calculator.
Questions to consider:
3. Explain why the values used in the Window setting create the appropriate graphical
display of two bars with the frequency of the 0s in the first and the frequency of the
1s in the second. In your response, explain why a y max of 350 is appropriate.
4. Determine the proportion of freshmen who will return to Chowan College next year.
5. For our problem we are interested in how much the proportion of freshmen returning
to Chowan College will vary from the expected 50%. Use the skills you have learned
to repeat the simulation of deciding the retention of 500 freshman many times.
Record the proportion of freshmen that continue on next year for each of the samples
of 500 trials. How many of the sample proportions fall within the interval you
predicted in question 2? Discuss why the results may or may not have been consistent
with your earlier prediction.
6. Recall that in our simulation, each freshman had a 50% chance of returning to school.
Did you get exactly 50% of the freshmen returning in the simulation? Discuss how
and why the empirical proportions from the simulation may have varied from 50%.
7. If we reduced the number of trials to 200 freshmen, what do you anticipate would
happen to the interval of proportions from the empirical data around the theoretical
probability of 50%? Why? Conduct a few samples with 200 trials and compare your
results with what you anticipated.
8. If we increased the number of trials to 999 freshmen, what do you anticipate would
happen to the interval of proportions from the empirical data around the theoretical
probability of 50%? Why? Conduct a few samples with 999 trials and compare your
results with what you anticipated.
9. Based on your experience with the simulations and the empirical data you collected,
what would be a reasonable interval for the proportion of freshmen that
administrators can expect to return to Chowan College for freshmen class sizes of
200, 500, and 999? Explain your predictions.
10. Discuss why it might be beneficial to have students simulate the freshman retention
Adapted from:
Lee, H. S., Hollebrands, K. F., & Wilson, P. H. (2010). Designing and using probability simulations. In Preparing to teach mathematics with
technology: An integrated approach to data analysis and probability (103-125). Dubuque: Kendall Hunt.
Tarr, J. E., Stohl Lee, H., & Rider, R. (2006). When data and chance collide: Drawing inferences from simulation data. In G. F. Burrill & P. C.
Elliott (Eds.), Thinking and reasoning with data and chance: Sixty-eighth NCTM yearbook (pp. 139-150). Reston, VA: National
Council of Teachers of Mathematics.
14
problem for several samples of sample size 500, as well as sample sizes of 200 and
999.
11. An important aspect of conducting a simulation of a repeated random event is to be
sure that students have a conceptual understanding of how the simulation and the
commands used in the computing tool represents the context of the problem. Consider
the following context:
Suppose a university gives away token spirit gifts to all incoming freshmen. As
they check in to pick up their class schedule, they get to randomly choose three
cards. Each card displays a different gift: keychain, window decal for a car, tshirt with college logo. Most freshmen would prefer the t-shirt.
Explain how you would help students use the graphing calculator to simulate this
context. Explicitly describe what the commands represent and how the students
should interpret the results.
12. Suppose that after conducting the above simulations, a student gets a result that a tshirt was chosen 20% of the time. The student is surprised and claims that this
proportion is too low from the expected 33.3%. What might this student be
misunderstanding? What are some questions you could pose and further simulations
you could suggest that might help this student?
Adapted from:
Lee, H. S., Hollebrands, K. F., & Wilson, P. H. (2010). Designing and using probability simulations. In Preparing to teach mathematics with
technology: An integrated approach to data analysis and probability (103-125). Dubuque: Kendall Hunt.
Tarr, J. E., Stohl Lee, H., & Rider, R. (2006). When data and chance collide: Drawing inferences from simulation data. In G. F. Burrill & P. C.
Elliott (Eds.), Thinking and reasoning with data and chance: Sixty-eighth NCTM yearbook (pp. 139-150). Reston, VA: National
Council of Teachers of Mathematics.