Drexel-SDP GK-12 ACTIVITY

Bayesian Penny TossSubject 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

SummaryIn 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 ConnectionThrough 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

KeywordsStatistics, modeling, prediction, classical, frequentist, Bayesian, data

Educational Standards PA State science: 3.27.B; 3.2.7.C; 3.2.7D

Pre-Requisite KnowledgeStudents should be aware of the basic concepts of probability.

Learning ObjectivesAt 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 ListEach group needs: Paper and pencils

Calculators

Computers

Introduction / MotivationScientists 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 / DefinitionsWord 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 IssuesNone

Investigating QuestionsFollow the worksheet.

AssessmentPost-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 ScalingNone

References

ContributorsJade Mitchell-Blackwood

CopyrightCopyright 2010 Drexel University GK-12 Program. Reproduction permission is granted for non-profit educational use.

Supporting Program: Drexel University GK12 Program Version: April 2010

Name: Date:

Bayesian Penny Toss

PART I -CLASSICAL PROBABILITY1. 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

Experimentation

1. Take 2 pennies and toss them. How many were heads? How many were tails? What is the probability of obtaining heads?

Make a chart in your notebook with 2 columns. Label one column “heads” and the other column “tails.”

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 PROBABILITYYou 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.