Download Pig Dice and Probability

Statistics and Probability for Middle-School Math Teachers:
Addressing the Common Core
Teacher Notes
Pig Dice and Probability
Logistics
This lesson is intended for students in Grade 7 as an introductory lesson to
the concepts of theoretical and empirical probabilities. Students will be
introduced to the idea of computing probabilities of a chance process
through observing frequencies of a desired event. They will also explore
the idea of a subjective probability when a theoretical probability cannot be
readily calculated.
Materials:
1 – set of dot dice per four students (or number cubes)
1 – set of pig dice per four students
Time: This lesson should be completed in two 45 – minute class periods.
Objectives/Standards
The objectives of this lesson are to:

Develop a probability model for a fair die and use it to find
probabilities of events related to rolling a fair die. Then, compare
the model to the results of an experiment and explain the possible
sources of the discrepancy CCSS.Math.Content.7.SP.7
References to Common Core are
adapted from NGA Center/CCSSO ©
Copyright 2010. National Governors
Association Center for Best Practices
and Council of Chief State School
Officers. All rights reserved.
CCSS.Math.Practice.2 CCSS.Math.Practice.4

Approximate the probability of a chance event of rolling a pig die
and obtaining a specific result by observing frequencies in the
data. Also, explore the idea of probability as long-run relative
frequencies. CCSS.Math.Content.7.SP.6 CCSS.Math.Content.7.SP.7
CCSS.Math.Practice.5

Summarize and describe the distributions of rolls of the standard
die and pig die. CCSS.Math.Content.6.SP.4 CCSS.Math.Content.6.SP.5
CCSS.Math.Practice.6
Introduction
For this inquiry, students will first explore the features of a standard, sixsided die and develop a uniform probability model to determine the
probability of a specific value of the die appearing on a roll. They will then
collect data to test their model and will examine the reasons why their
observed frequencies may not match the theoretical model they initially
proposed. Then, students will attempt to determine a theoretical model for
a die in the shape of a pig. Due to the unusual shape of the pig dice, it is
very difficult to determine the theoretical probability of any particular pig
position. In fact, according to John C. Kern of Duquesne University, exact
probabilities are unknown. Thus, students will reason that long-run
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Statistics and Probability for Middle School Teachers: Addressing the Common Core
Professional Learning Day; February 28, 2014
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Statistics and Probability for Middle-School Math Teachers:
Addressing the Common Core
Teacher Notes
relative frequencies must be used for this particular die to determine an
approximation of the probability for a specific pig position to occur.
Students should be grouped into teams of four students arranged in a
square/rectangular pattern and then divided further into partner teams
on the same side of the table (shoulder partners). For the first two
activities in this lesson, students will work with their partner teams.
Activity 1: Investigating a standard, six-sided die
Is a standard, six-sided die “fair”?
For this activity, students will:


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Examine the shape of the die to determine the “sample space” or
list of possible outcomes if the die were rolled.
Use the idea of the symmetric shape of the die to determine the
probability of a specific value occurring on any given roll.
Determine the theoretical probabilities for each possible outcome in
the sample space.
Develop a data collection strategy to test their theoretical model.
Investigate their empirical data to determine if the data does/does
not support their theoretical model and infer reasons why this
occurred. Students may suggest changes to their models to reflect
their changing ideas about long-run relative frequencies.
Students will first examine the shape of a standard die and describe their
observations. Sample student responses could be as follows:

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The die is a cube.
There are six equal sides that are squares.
There are values of 1, 2, 3, 4, 5, and 6 on the die (or dots if using
dot dice).
The die is a uniform weight – one side is not heavier than the other
side.
From these observations, students should then roll the die a few times to
see the possible outcomes of each roll. From there, they can list the
sample space of a single, six-sided die as 1, 2, 3, 4, 5 and 6.
Assuming that every roll is independent of the previous roll, students can
determine the theoretical probability of obtaining a specific value on a roll
to be the uniform distribution of 1 desired outcome/ 6 possible
outcomes. Upon completing the table in the student pages, they should
conclude that the likelihood for any number being rolled is the same
(uniform). Thus, they can create a table of theoretical probabilities as
follows:
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Statistics and Probability for Middle School Teachers: Addressing the Common Core
Professional Learning Day; February 28, 2014
Statistics and Probability for Middle-School Math Teachers:
Addressing the Common Core
Teacher Notes
Theoretical Probabilities of a roll of a Six-sided
die
Outcome
1
2
3
4
5
6
Probability
1
6
1
6
1
6
1
6
1
6
1
6
Now, student teams will collect data on 25 trials of rolling the die and
recording the outcome of the roll. From these data, students will then
calculate their experimental probabilities using the following formula:
𝑃(𝐸𝑣𝑒𝑛𝑡 𝐴) =
# 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 𝑤𝑎𝑠 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠
Following the data collection activity, students will be asked to compare
their experimental data table to the theoretical table they generated and
discuss the similarities/differences in the data. Ask students to share
their observations. They will also share their data with other groups to
observe the variability in the recorded probabilities. From there, groups
will develop a hypothesis on why they believe these differences are
occurring and will design an experiment that they believe will give them
data that more closely approximates the theoretical values that were
calculated using the uniform probability distribution. Whole class
discussion will focus on the sharing of student observations and
conjectures. The following questions may be used to generate
conversation:
 Did the experimental data you recorded exactly match the uniform
distribution of 1/6 for each outcome on the first table?
 What reason(s) might there be for the differences you found, if
indeed you found any?
 Was your experimental table exactly the same as other groups in
the class?
 What could explain the variability from group to group?
 Do you think that your die is a fair die? Why or why not?
 What would it take for you to be convinced that your die was fair?
 How could you design an experiment to get closer to the actual
theoretical frequencies of 1/6 for each value on the die?
Following discussion, students may redesign their plans and then carry
out their revised plans focusing on increasing the number of trials. They
will then compare their new experimental data again to the theoretical
data and their first set of data to determine what effect increasing the
number of trials has on the data. This leads to a definition of probability
as a long-run relative frequency.
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Statistics and Probability for Middle School Teachers: Addressing the Common Core
Professional Learning Day; February 28, 2014
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Statistics and Probability for Middle-School Math Teachers:
Addressing the Common Core
Teacher Notes
Activity 2: Investigating pig dice
What is the probability of landing in a specific pig position with a single
toss of the pig die?
In the next part of this lesson, students will look at a die in the shape of a
pig and will investigate the possible outcomes when this die is tossed.
Students should rationalize that the geometry of the pig die makes it
difficult, if not impossible, to theoretically calculate the likelihood of a
particular pig position from occurring. They would need to generate
subjective probabilities about their personal knowledge of what they
believe is most likely to occur. For example, the pig may land on its side
more often since this is the “largest” part of the pig die. Likewise, the pig
landing on its back may occur more often than the pig resting on its ear.
Following the completion of Page 6 of the student pages, pose the
following questions to the class:
 What pig positions are possible with a single pig die?
 If there are six different pig positions, do you believe they are
equally likely to occur as is the case with the standard die? Why or
why not?
 What would you predict the probabilities of the different pig
positions to be? What specifically about the pig die would lead you
to your predictions?
 How could we attempt to determine the likelihood of a specific pig
position? What things might you have to consider?
Following the debrief on the initial section of this activity, students will
formulate a plan that would allow them to calculate the experimental
probabilities for each pig position. Allow partner teams time to formulate
their plans, and then share their ideas. The class will decide, with the
guidance of the teacher, a particular plan or combination of individual
plans to follow. For the initial data collection, guide students to choose a
plan in which the number of pig tosses is at least 25; data is recorded in a
table listing the pig position that occurred at each toss or trial. Students
may also take turns within their partner teams. This will generate
additional data sets to use in the next part of the activity.
Following the data collection phase, each partner team will calculate the
relative frequencies, or experimental probabilities, for each pig position by
using the following formula:
𝑃(𝑃𝑖𝑔 𝑃𝑜𝑠𝑖𝑡𝑖𝑜𝑛) =
# 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑤𝑎𝑠 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠
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Statistics and Probability for Middle School Teachers: Addressing the Common Core
Professional Learning Day; February 28, 2014
Statistics and Probability for Middle-School Math Teachers:
Addressing the Common Core
Teacher Notes
Students can adapt the formula from Activity 1 to calculate the
experimental probabilities for the pig positions.
Students should realize that they have a relatively small number of trials,
so the confidence in their calculated probabilities should be quite low.
One way to increase the number of trials without tossing the pig die more
times, which may be impractical given time constraints, is to consolidate
the class data. Student partner teams are now asked to gather class data
for a particular pig position which they are assigned. There are six
positions, so the pig positions may be analyzed by more than one group.
This is desirable since students are not instructed on what type of
graphical display to use. The most likely displays will be either a dot plot
or a boxplot to represent the frequency of the assigned pig position.
Frequency of Razorbacks Tossed (per 25 tosses)
3
6
9
12
Razorback (per 25 tosses)
15
Variable
Mean Median
Razorback (per 25 tosses) 7.800
7.500
Created using Minitab software
Number of Razorbacks
The above display
shows an example of a dot plot depicting the frequency
of razorbacks observed per 25 tosses of the pig die. Students could then
revise their probability of a razorback to 7.8/25 or .312. Alternatively,
they could add up the total number of razorbacks observed (234) and
divide by the total number of tosses (750).
**Please note that revised probabilities in the table should sum to 1!**
The enduring understanding that students should take away from this part
of the lesson is that not every outcome can be assigned a theoretical
probability, and often the best we can do is an empirical probability that
depends upon the number of trials.
Pig dice images from: http://www.softcom.net/users/mikey719/pass_the_pigs/PassThePigs.html
Activity 3: Putting it all together
For this final part of the lesson, students will again be working with their
partners in order to summarize the concepts that were explored in the first
two activities. Initially they will work with their partner to check the box
which indicates if they believe the statement is true for the standard die,
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Statistics and Probability for Middle School Teachers: Addressing the Common Core
Professional Learning Day; February 28, 2014
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Statistics and Probability for Middle-School Math Teachers:
Addressing the Common Core
Teacher Notes
the pig die, both, or neither. Then, student teams may compare their
answers with the other team at their table and discuss the items where
they do not have consensus. Following this discussion, students will write
a final summary of the main ideas they learned in this lesson. These
ideas will be debriefed in a whole-class discussion. The following will
serve as a guideline for facilitating discussion on this part of the activity:
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The sum of the probabilities for all possible outcomes in a sample
space is 1.
Both theoretical probability and empirical probability are ratios.
A uniform probability distribution has all outcomes equally likely. A
chance process that is uniform is considered “fair”.
Since theoretical probability is based on a calculation of what is
expected to occur given all possible outcomes, it does not always
accurately predict what does occur in a given trial or series of trials.
Experimental probability is an approximation of a theoretical
probability and tends to approach the theoretical probability after a
large number of trials.
In cases where the theoretical probability cannot be calculated, we
need to rely on experimental probability.
Conclusion
This lesson is intended to allow students the time and opportunity to
explore the concepts of theoretical and empirical probability. Furthermore,
students will investigate the idea of fairness as equally likely and will look
at the effect of long-run relative frequencies as an approximation of the
probability of a chance event. Using a guided inquiry model, students are
asked to explain their reasoning, justify their thoughts, and predict
outcomes based on both prior knowledge and what they are learning
throughout the inquiry. The Common Core Mathematics Practices, as
outlined in the Common Core State Standards Initiative, are integrated
throughout the lesson. Students made sense of the tasks they were
assigned and analyzed relationships between their calculated probabilities
for the standard die and experimental (MP.1) and revised their thinking as
new information was presented. They constructed a uniform probability
model to describe the outcomes of rolling a standard die (MP.4) and used
a similar quantitative relationship to define the experimental probabilities
for both the pig die and the standard die (MP.3). Students looked at
patterns in the data to discover that the sum of the probabilities for a
sample space was 1 (MP.7) and used the models obtained through
experimentation with manipulatives (MP.5) to communicate their findings
precisely through written and graphical formats (MP.6).
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Statistics and Probability for Middle School Teachers: Addressing the Common Core
Professional Learning Day; February 28, 2014
Pig Dice and Probability
Name: _________________________________ Partner Name: ____________________________
For this inquiry, you and a partner will investigate a standard die or number cube and
a die in the shape of a pig and determine the likelihood of various rolls occurring for
each of the different-shaped dice.
Activity 1: The Standard Die
Investigate a standard die or number cube. In the space below, write
down all observations you and your partner make about the number
cube. Be ready to share your observations with your classmates!
Roll the die or number cube several times. What observations can you make about
the results of your rolls?
Based on your observations, do you believe this is a “fair” die or number cube?
Explain your reasoning.
Pig Dice and Probability Student Pages
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If we can consider the die to be “fair”, then each value on the die has an equal
likelihood, or chance, of occurring. So, to figure out what the probability would be,
we first need to list all of the possible outcomes of rolling a single six-sided die. We
call this list the “sample space.” In the space below, list the possible outcomes of
rolling a single die or a number cube:
Now, to calculate the probability for a specific value on the die, say a roll of the
number 5, the following formula would be used:
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑟𝑜𝑙𝑙𝑖𝑛𝑔 𝑎 (5) =
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑑𝑖𝑒 𝑡ℎ𝑎𝑡 𝑎𝑟𝑒 𝑎 5
𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
Use this idea to complete the following table:
Probability for a specific roll of the die
Outcome
Probability
What can you conclude from looking at the table above?
Now, you and your partner are going to test whether or not the theoretical
probabilities you generated in your table are actually a good model for what
happens if you roll a single, six-sided die. For this first test, you are going to roll the
die 25 times and record in a table on the next page the results of your tosses.
Pig Dice and Probability Student Pages
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2
Results of Rolling a Single Six-Sided Number Cube
Trial Value Trial Value Trial Value Trial Value Trial Value
# on die # on die # on die # on die # on die
1
6
11
16
21
2
7
12
17
22
3
8
13
18
23
4
9
14
19
24
5
10
15
20
25
Let us consider rolling the die a random experiment. Then, let A be an outcome of
the random experiment (a particular roll of the die) . Then A is called an event
(similar to an outcome). The experimental probability of the event A is given by the
following formula:
𝑃 (𝐴) =
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 𝑜𝑐𝑐𝑢𝑟𝑠
𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠
where P(A) represents the probability of event A occuring. Experimental probability
is also known as the relative frequency definition of probability. Using the above
definition, complete the following table:
Experimental Probability for a specific roll of the die
Event
Probability
Now, compare the probabilities in the table above for each value on the die to the
ones in the previous table. On the next page, discuss the similarities and
differences in the probabilities for the two tables.
Pig Dice and Probability Student Pages
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Similarities and Differences Between the Theoretical Probability Distribution
(first table) and the Experimental Probability Distribution (second table):
Share your experimental data with other groups. Write down some observations you
make as you compare your experimental probabilities to those of your classmates.
Based on your experimental data, is the six-sided die fair?
With your partner, design another experiment that might give you data that better
approximates the theoretical probabilities of rolling a six-sided die.
Be ready to share your observations and design with your classmates!
Pig Dice and Probability Student Pages
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4
Carry out your experiment in the space below. Record any necessary data and
make any necessary calculations.
What conclusion(s) can you form based upon your revised experiment?
Pig Dice and Probability Student Pages
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Activity 2: The Non-standard Pig Die
Given a die in the shape of a pig, toss the pig die several times to
determine the ways that the pig die can land on the table. Record your
observations below:
Compare your list above with the other team at your table. Add any additional “pig
positions” to the list.
Do you believe that all of the different pig positions are equally likely? In other
words, is the pig die a “fair” die? Justify your response.
Based on the shape of the pig die, is it possible to calculate the probability of each
of the different pig positions shown. If yes, describe how you might go about
calculating the probabilities. If no, describe why it would be impossible.
Be prepared to share your responses with the class.
Pig Dice and Probability Student Pages
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For the following experiment, we are going to assume the following pig positions:
Pink sider
Dot sider
Razorback
Trotter
Snouter
Leaning Jowler
With your partner, create a plan for determining the probability of rolling each
specific pig position. We will be deciding as a class which plan to use to conduct
the experiment.
Carry out the plan decided upon by the class. Record all necessary information.
Pig Dice and Probability Student Pages
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Based on your data, complete the following table:
Experimental Probability for a specific roll of the pig die
Event
Pink sider
Dot sider
Razorback
Trotter
Snouter
Leaning
Jowler
Probability
How confident are you in the results of your experiment?
What could be done to increase the confidence in the probabilities for the pig
positions?
Your partner team will be assigned a particular pig position. Using the class data,
you will create a graphical display for the observed frequency of the assigned pig
position for each team on a sheet of paper provided to you by your teacher. Your
display should include the following information.
 Title (Pig Position Investigated)
 Graphical Display
 Numeric Summary of key features of graphical display
Based on the class data, revise your probabilities for each of the pig positions.
Revised Experimental Probability for a specific roll of the pig die
Event
Pink sider
Dot sider
Razorback
Trotter
Snouter
Leaning
Jowler
Probability
What conclusions can you make based on the above table?
Pig Dice and Probability Student Pages
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Activity 3: Putting it all together
In the following activity, you will compare and summarize the two
experiments that were conducted using both the six-sided die and the
pig die. For the first part of this activity, place a check mark in the box
if the statement applies to the experiment with the standard die or the
pig die. Both boxes may be checked if the statement applies to both types of die, or
neither box may be checked if the statement does not apply to both types of die.
Statement
It is possible to calculate a theoretical probability for
determining the likelihood of a specific roll of the die.
It is possible to calculate an empirical (experimental)
probability for determining the likelihood of a specific roll of
the die.
The sum of the probabilities for all of the possible outcomes
for the die was always 1.
All of the groups in my class had the same results when
tossing the die a fixed number of trials.
There was variability in the experimental probabilities from
group to group when tossing the die a fixed number of trials.
All of the possible outcomes are equally likely.
The probability can only be approximated from the observed
frequencies.
Increasing the number of tosses of the die (trials) results in
the experimental probability more closely approximating the
theoretical probability.
The die is a “fair” die.
Based on the above table, what key ideas can you conclude about the probability of
a chance event?
Pig Dice and Probability Student Pages
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