Drexel-SDP GK-12 ACTIVITY
Bayesian Penny Toss
Subject Area(s):
problem solving, reasoning & proof
Activity Title :
Bayesian Penny Toss
Grade Level:
6 (5-12)
Time Required:
45 minutes
Group Size:
2
Expendable Cost per Group US$0
Summary
In this activity students will conduct several trials of penny tosses to gain an understanding of the
three distinct types of probability: classical, frequentist and Bayesian. While this lesson focuses
on statistics primarily, it is equally relevant to engineering research. This activity helps students
make the connection between science and engineering, as it promotes an understanding of how
data is used to make predictions about real world phenomena that have not actually occurred, but
need to be taken into account when engineering designs are implemented.
Engineering Connection
Through other activities the students have been exposed to design and experimentation. They are
aware of how engineers and scientists use experimentation, but may not know exactly how the
data is used after experiments are performed. It may also be clear to them that all experiments
under ever condition cannot be performed in the laboratory. As a result of this, engineers need to
make predictions about things that have not occurred yet. Statistics is the tool used to accomplish
this. This introduction to statistical inference is a preview into how data is used by engineers to
model a wide variety of natural and engineered phenomena. Modeling and statistical inference
help engineers make predictions about the usefulness and performance of their designs in real
world situations.
Engineering Category
(#3) Connects science to engineering
Keywords
Statistics, modeling, prediction, classical, frequentist, Bayesian, data
Educational Standards
PA State science: 3.27.B; 3.2.7.C; 3.2.7D
Pre-Requisite Knowledge
Students should be aware of the basic concepts of probability.
Learning Objectives
At the end of this lesson students will be able to (1) describe the three different types of
probability using a concrete example and (2) describe how engineers use statistics in their
research.
Materials List
Each group needs:
 Paper and pencils

Calculators

Computers
Introduction / Motivation
Scientists and engineers have to make predictions about future events and the performance of
their designs on a regular basis. Both groups test outcomes using experiments, but it is
impossible and costly to test under every condition, so there is always some remaining
uncertainty even after experimentation. Statistics are used in engineering and scientific research
to help explain how certain we are about future events or predictions based on the results of our
experiments and/or our prior knowledge. Statistics is the branch of mathematics that deals with
the collection, organization, analysis, and interpretation of numerical data. In order to accomplish
this, statistics uses mathematical theories of probability. Statistics is especially useful in drawing
general conclusions about a set of data from a sample of the data. There are three different
interpretations of probability that can be used: Classsical, Frequentist and Bayesian. Today we
will explore the difference between them
Vocabulary / Definitions
Word
Definition
probability the relative possibility that an event will occur, as expressed by the ratio of the
number of actual occurrences to the total number of possible occurrences
statistics
the branch of mathematics that deals with the collection, organization, analysis,
and interpretation of numerical data. Statistics is especially useful in drawing
general conclusions about a set of data from a sample of the data.
Bayesian
pertaining to statistical methods based on Thomas Bayes' probability theorem
involving prior knowledge and accumulated experience
modeling
the representation, often mathematical, of a process, concept, or operation of a
system, often implemented by a computer program
Procedure
With the Students
1. Give each student a copy of the worksheet, “Bayesian Penny Toss.”
2. Hand out 20 pennies to each group of 2 students.
3. Let the students work through Part 1 of the worksheet, then break for a discussion.
4. Ask them investigating questions from the worksheet. Since the concept from one Part
leads into an understanding of the subsequent sections, it is important to discuss them as
they are completed.
5. Continue to Parts 2 and 3, breaking for discussion in between sessions.
6. Conclude with the final assessment.
Safety Issues
None
Investigating Questions
Follow the worksheet.
Assessment
Post-Activity Assessment
Have students describe how this knowledge could be applied to a real world situation or
engineering project. Their statement should include all 3 types of probabilities as they apply to
the situation or problem.
Activity Scaling
None
References
Contributors
Jade Mitchell-Blackwood
Copyright
Copyright 2010 Drexel University GK-12 Program. Reproduction permission is granted for nonprofit educational use.
Supporting Program: Drexel University GK12 Program
Version: April 2010
Name:
Date:
Bayesian Penny Toss
PART I -CLASSICAL PROBABILITY
1. Look at a penny. If you flip the coin, what is the probability of obtaining heads
or tails?
The probability of obtaining heads or tails is 50:50 or 50% because there is a 1 in 2
chance that it will be heads or tails.
Can you think of any other phenomena that have similar probabilities?
How about whether a child will be a girl or boy? The probability of a baby girl or
boy is also 50:50 or 50%. But, what if a family has 2 children? Will 50% always be
girls and 50% always be boys? What if the family had 4 children or 6 or 10? The
answer is “Not necessarily!” In order to answer this question and many others we
use “frequentist” statistics.
Scientists and engineers have to make predictions about future events and the
performance of their designs on a regular basis. Both test outcomes using
experiments, but it is impossible and costly to test under every condition, so
there is always some remaining uncertainty even after experimentation.
PART II - FREQUENTIST PROBABILITY
Make a chart in your notebook with 2 columns. Label one
column “heads” and the other column “tails.”
Experimentation
1.
Take 2 pennies and toss them. How many were heads? How many were
tails? What is the probability of obtaining heads?
Repeat. Is the probability of obtaining heads or tails the same?
Repeat 18 more times. Record the probability each time in your chart.
Total the values you have in each column. Are they equal?
2.
Now let’s try it with 4 pennies. Take 4 pennies and toss them. How many
were heads? How many were tails? What is the probability of obtaining
heads?
Repeat. Is the probability the same?
Repeat 8 more times. Record the probability each time. Total the values
you have in each column. Are they equal?
You could continue this process with more and more pennies if you wish.
PART III – BAYESIAN PROBABILITY
You may have decided to continue experimenting with more pennies, but at
some point you will have to stop. This is the case for all engineering and scientific
experimentation.
One of the ways that you can make predictions about future events is through
Bayesian statistics. “Bayesian” statistics is a “measure of a state of knowledge.”
A Bayesian probabilist specifies a prior probability (sometimes based on prior
experimentation), then updates that probability in light of new data. So, in this
case you know that the probability of obtaining heads or tails is about 50% every
time you toss the coin. This is your prior probability. You may not have obtained
this result in you experiments though. You could combine this information using a
formula called Bayes’ theorem.