Transcript “Counting Counts”

A Workshop for Grade 5-12 Teachers:
Probability Simulations Based on
Adaptations of National Science
Foundation Sponsored
Curriculum Materials for University
Probability and Statistics
Tracy Goodson-Espy,
Appalachian State University
Ginger Holmes Rowell,
Middle Tennessee State University
M. Leigh Lunsford,
Longwood University
Presentation to the Society for Information
Technology and Teacher Education
International Conference
Phoenix, AZ March 1-5, 2005
Project Objectives
 To improve post-calculus students
learning of probability & statistics.
 To provide students with better
preparation for their future careers in
mathematics & statistics, mathematics
education, and computer science.
 To adapt the project materials, where
appropriate for use in grades 5-12.
*This project was partially supported by the National Science Foundation.
The project started in June 2002 and concluded in August 2004.
The Materials for A&I
 A Data-Oriented, Active Learning, PostCalculus Introduction to Statistical
Concepts, Methods, and Theory (SCMT)
A. Rossman, B. Chance, K. Ballman
NSF DUE-9950476
 Virtual Laboratories in Probability and
Statistics (VLPS)
K. Siegrist - NSF DUE-9652870
A Collaborative Approach
A&I Materials into Post Calculus Prob/Stat Courses
Athens State Univ./
Univ. of Alabama,
Huntsville
M. Leigh Lunsford
Middle Tenn. St. Univ.
Ginger Holmes Rowell
Univ. of Alabama, Huntsville
Tracy Goodson-Espy
Provide Objective Independent Assessment of A&I
NSF DUE-0126401
Courses for A&I
Athens State Univ:
•
Applied Statistics & Probability I (3 hrs)
Clientele: CS, Math, Math Ed. Majors - Prereq: Calculus II
UAH:
•
Introduction to Probability (3 hrs)
Clientele: Engineering, CS, Math, Math Ed - Prereq: Calculus II
•
Introduction to Mathematical Statistics (3 hrs)
Clientele: Math & Math Ed. Majors - Prereq: Intro to Prob. & Cal III
MTSU:
•
Probability & Statistics (3 hrs)
Clientele: CS, Math. Ed. Majors - Prereq: Calculus I
•
Data Analysis (1 hr)
Clientele: CS, Math. Ed. Majors - Prereq: Calculus I
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Statistical Concepts, Methods, and
Theory (SCMT): A Small Sample of
Materials
Activity
Context
Concepts
Description
Friendly
Observers
Randomization,
simulation, p-value
Uses cards (23 per student) and Minitab to
simulate a randomization test to estimate pvalue from a 2x2 table for a psychology study.
Equal
Likeliness
Random
Babies
Sample space, long-run
relative frequency,
random variable,
expected value,
simulation
Uses index cards (4 per student) and Minitab to
simulate the matching problem and develops
probability calculations with equally likely
outcomes.
Fishers
Exact Test
Friendly
Observers
Counting rules,
hypergeometric
probabilities
Exact probabilities for simulation in
Randomization Test
General vs.
Specific
The Birthday
Problem
Applications of counting
techniques, complement
rule
Does calculations for the birthday problem
(using a spreadsheet) contrasting any birthday
vs. a specific birthday
Probability
Rules
100 top films,
2000 Michigan
primary
Variety of basic
probability rules
Discovery approach through two-way tables and
some Venn diagrams. HW is very interesting
Randomization Test
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Virtual Laboratories in Probability &
Statistics: An Example
Games of Chance
Contents
1. Poker
2. Poker Dice and Chuck-a-Luck
3. Craps
4. Introduction
5. Roulette
6. The Monty Hall Problem
7. Lotteries
8. Notes
Applets
•
Poker Experiment
•
Poker Dice Experiment
•
Chuck-a-Luck Experiment
•
Craps Experiment
•
Roulette Experiment
•
Monty Hall Game
•
Poker Experiment Applet
Monty Hall Experiment
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Examples of 5-12 Topics
 Basic probability rules
 Types of probability
 Classical/theoretical
 Relative
frequency/experimental
 Fundamentals of counting
 Permutations
 Combinations
 Multiplication
principle
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Teacher Workshops &
Presentations
 Modes:
 On-site
at MTSU and at Huntsville HS
 Via distance learning to middle Tennessee
counties and to Huntsville, AL City Schools
 Alabama Council of Teachers of
Mathematics, October 2003
 North Carolina Council of Teachers of
Mathematics Meeting, October 2004
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Teacher Workshops
 Topics:
 Probability
Review and Counting
Fundamentals
 Activity-Based Probability and Statistics
 Example Activity Follows: The Random
CDs activity was modified from an activity
in SCMT
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Random CDs
 Topics and Goals
Explore
the idea of equally likely
probabilities
Compare theoretical and relative
frequency probabilities
Develop
an understanding of expected
value as the long-run average value
achieved by a random process
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Random CDs
 Activity - Random CDs
 Materials
4
index cards & sheet of paper
Web-based Applet Simulation (from VPLS)
Computer with Minitab (optional)
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Random CDs
 Scenario - 4 music CDs are returned
at random to 4 jewel boxes
 Exploration - Use simulation to
determine what will happen in the long
run
 Practice - enumeration to find exact
probability
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CD Artist Titles
Beatles
Springsteen
U2
Rolling Stones
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Randomly Matching CDs &
Jewel Cases
Repetition
Number
1
2
3
4
5
Number
of
Matches
0
2
1
1
1
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Example Results Seen in the
October 2004 NCCTM Workshop
Matches
Trial 1
2
3
4
5
Proportion
(count/total)
0
10
8
12
8
7
45/150 = 0.3
1
9
7
6
9
10
41/150 = .273
2
8
7
7
8
6
36/150 = .24
4
3
8
5
5
7
28/150 = .187
Total
30
30
30
30
30
1.00
Average
1.23*
1.76
1.33
1.5
1.67
*Average: (10(0) + 9(1) + 8(2) + 0(3) + 3(4) )/30=
37/30 = 1.23
Sample Space Enumeration
(# of Matches)
1 2 3 4 (4) 2 1 3 4 (2) 3 1 4 2 (0) 4 1 2 3 (0)
1 2 4 3 (2) 2 1 4 3 (0) 3 1 2 4 (1) 4 1 3 2 (1)
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Sample Space Enumeration
(# of Matches-each one assumed to
be equally likely)
1 2 3 4 (4) 2 1 3 4 (2) 3 1 4 2 (0) 4 1 2 3 (0)
1 2 4 3 (2) 2 1 4 3 (0) 3 1 2 4 (1) 4 1 3 2 (1)
1 3 2 4 (2) 2 3 1 4 (1) 3 2 1 4 (2) 4 2 1 3 (1)
1 3 4 2 (1) 2 3 4 1 (0) 3 2 4 1 (1) 4 2 3 1 (2)
1 4 2 3 (1) 2 4 1 3 (0) 3 4 1 2 (0) 4 3 1 2 (0)
1 4 3 2 (2) 2 4 3 1 (1) 3 4 2 1 (0) 4 3 2 1 (0)
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Tree Diagrams
1st CD
2nd CD
2
1
3
4
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3rd CD
3
4
2
4
2
3
4th CD
4
3
4
2
3
2
Fundamentals of Counting
 Multiplication Principle:
 If there are a ways of choosing one
thing, b ways of choosing a second thing
after the first is chosen, and c ways of
choosing a third thing after the first two
have been chosen…and z ways of
choosing the last item after the earlier
choices, then the total number of choice
patterns is a x b x c x … x z
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CD Matches Example
Continued: Exact Probabilities
P(0 matches) = 9/24 = 3/8=.375
P(1 match) = 8/24=1/3=.333
P(2 matches) = 6/24=1/4=.25
P(3 matches) = 0/24=0
P(4 matches) = 1/24=.042
How do these compare to our class
empirical
estimates
from
the
simulation?
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Comparing Experimental Probabilities
to Theoretical Probabilities
Number of
Matches
0
1
2
3
4
Experimental
Probability
(150 trials)
.3
.273
.24
0
.187
Theoretical
.375
.333
.25
0
.041667
Probability
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Comparing Experimental Probabilities
to Theoretical Probabilities
 In this instance of the simulation,
the experimental and theoretical
results are rather closer for 0 and 2
matches but not as close for 1 and
4 matches.
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Need for Computer Simulation to
Support Hands-on Activity
 In a case such as this one, students could reach

erroneous conclusions about the relationship
between relative frequency and theoretical
probabilities.
Without the computer simulation aspect of the
activity, it is easy for students to think something
“went wrong” in a particular instance of the
classroom experiment and to remain completely
unaware of the real point of the lesson.
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Minitab or VLPS Applet
Simulation
# of Matches
0
1
2
3
4
mean
10
repetitions
.2 (2/10)
.5 (5/10)
.2 (/10)
0
.1 (1/10)
1.3
100
repetitions
.34
.38
.22
0
.6
1.06
1000
repetitions
.356
.348
.256
0
.0406
1.02
10,000
repetitions
.3732
.3376
.2485
0
.0407
1
Theoretical
.375
.33333
.25
0
.041667
1.00
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Expected Value
The long-run average value achieved
by a numerical random process is
called the expected value of the
random variable. To calculate this
expected value from the exact
probability distribution, multiply each
outcome of the random variable by its
probability, and then add these up
over all possible outcomes.
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Expected Value
P(0 matches) = 9/24
P(1 match) = 8/24
P(2 matches) = 6/24
P(3 matches) = 0/24
P(4 matches) = 1/24
(0)(9/24)=0
(1)(8/24)=8/24
(2)(6/24)=12/24
(3)(0/24)=0
(4)(1/24)=4/24
8/24 + 12/24+ 4/24 = 1
We can see that the larger the number of repetitions
of the experiment, the closer the experimental
average (or sample mean) becomes to the expected
value of the random variable (or distribution mean).
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Conclusions
 Tactile and computer/online
simulations should be used in
tandem to insure that students
understand the concepts involved
and so that they do not develop
common misconceptions.
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Contact Information
 Tracy Goodson-Espy, ASU
[email protected]
 Ginger Holmes Rowell, MTSU
[email protected]
 M. Leigh Lunsford, Longwood Univ.
[email protected]
NSF DUE-0126401
Collaborative Project
Website
 Probability & Statistics
Activities
to download
NCTM standards information
Links with teaching resources
 http://www.mathspace.com/NSF_
ProbStat/index.htm
NSF DUE-0126401