9.4.3B Simulations
Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes.
Understand that the Law of Large Numbers expresses a relationship between the probabilities in a probability model and the experimental probabilities found by performing simulations or experiments involving the model.
Use random numbers generated by a calculator or a spreadsheet, or taken from a table, to perform probability simulations and to introduce fairness into decision making.
For example: If a group of students needs to fairly select one of its members to lead a discussion, they can use a random number to determine the selection.
Overview
Life is a school of probability. - Walter Bagehot
Students have calculated probabilities to solve real world problems in the form of experimental probabilities, probability as a fraction of sample space or area, and used random number generators to conduct simulations. This standard builds on this knowledge in that students learn counting strategies and more advanced probability concepts such as intersections, unions, complements, and conditional probability. Students work on "formalizing probability procedures and language, creating and interpreting probability distributions to solve real world problems, implementing simulation techniques, and presenting cohesive arguments in oral and written form." (Minnesota Math Frameworks, 1997, Randomness and Uncertainty, P. 19.) Students investigate probability problems with technology through simulations. Virtually all jobs require decision-making capabilities under uncertain conditions. Technology enables large amounts of data to be analyzed, but it is humans who must make sense of the data.
All Standard Benchmarks
9.4.3.1 Select and apply counting procedures, such as the multiplication and addition principles and tree diagrams, to determine the size of a sample space (the number of possible outcomes) and to calculate probabilities.
For example: If one girl and one boy are picked at random from a class with 20 girls and 15 boys, there are 20 × 15 = 300 different possibilities, so the probability that a particular girl is chosen together with a particular boy is 1/300.
9.4.3.2 Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes.
9.4.3.3 Understand that the Law of Large Numbers expresses a relationship between the probabilities in a probability model and the experimental probabilities found by performing simulations or experiments involving the model.
9.4.3.4 Use random numbers generated by a calculator or a spreadsheet, or taken from a table, to perform probability simulations and to introduce fairness into decision making.
For example: If a group of students needs to fairly select one of its members to lead a discussion, they can use a random number to determine the selection.
9.4.3.5 Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems.
For example: The probability of tossing at least one head when flipping a fair coin three times can be calculated by looking at the complement of this event (flipping three tails in a row).
9.4.3.6 Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets.
9.4.3.7 Understand and use simple probability formulas involving intersections, unions and complements of events.
For example: If the probability of an event is p, then the probability of the complement of an event is 1 - p; the probability of the intersection of two independent events is the product of their probabilities. Another example: The probability of the union of two events equals the sum of the probabilities of the two individual events minus the probability of the intersection of the events.
9.4.3.8 Apply probability concepts to real-world situations to make informed decisions.
For example: Explain why a hockey coach might decide near the end of the game to pull the goalie to add another forward position player if the team is behind. Another example: Consider the role that probabilities play in health care decisions, such as deciding between having eye surgery and wearing glasses.
9.4.3.9 Use the relationship between conditional probabilities and relative frequencies in contingency tables.
For example: A table that displays percentages relating gender (male or female) and handedness (right-handed or left-handed) can be used to determine the conditional probability of being left-handed, given that the gender is male.
Benchmark Group B
9.4.3.2 Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes.
9.4.3.3 Understand that the Law of Large Numbers expresses a relationship between the probabilities in a probability model and the experimental probabilities found by performing simulations or experiments involving the model.
9.4.3.4 Use random numbers generated by a calculator or a spreadsheet, or taken from a table, to perform probability simulations and to introduce fairness into decision making. For example: If a group of students needs to fairly select one of its members to lead a discussion, they can use a random number to determine the selection.
What students should know and be able to do [at a mastery level] related to these benchmarks
- Students should be able to design and conduct an experiment or simulation using repeated trials and analyze the results by comparing them to theoretical probabilities.
- Students should have multiple experiences to understand that the law of large numbers means that the average of the results obtained from a large number of trials should be close to the probabilities in the probability model.
- They should use multiple representations to solve probability problems.
Work from previous grades that supports this new learning includes:
- Students have had experiences modeling different real world probabilistic situations and representing probabilities using fractions, decimals, and percents.
- Students should be familiar with using the 0-1 scale as a measure of probability.
- Students have worked with simulations that highlight relative frequency and theoretical probability.
- Students can conduct a simple simulation using hands-on techniques or technology
- Students have worked with reading, writing, and comparing rational numbers expressed as integers, fractions, and decimals.
NCTM Standards
Data Analysis and Probability Standards
3. develop and evaluate inferences and predictions that are based on data
- understand how sample statistics reflect the values of population parameters and use sampling distributions as the basis for informal inference
4. understand and apply basic concepts of probability
- understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases
- use simulations to construct empirical probability distributions
- compute and interpret the expected value of random variables in simple cases
- understand the concepts of conditional probability and independent events
- understand how to compute the probability of a compound event.
Common Core State Standards (CCSS)
S-IC: Making Inferences and Justifying Conclusions
Understand and evaluate random processes underlying statistical experiments
2. Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?
S-CP: Conditional Probability and the rules of Probability
Understand independence and conditional probability and use them to interpret data
1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer
Use the rules of probability to compute probabilities of compound events in a uniform probability model
6. Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.
7. Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.
8. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems.
S-MD: Using Probability to make decisions
Calculate expected values and use them to solve problems
1. (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.
2. (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
3. (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.
4. (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?
Use probability to evaluate outcomes of decisions
5. (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast- food restaurant.
b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.
6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
S-ID: Interpreting Categorical and Quantitative data
Summarize, represent, and interpret data on two categorical and quantitative variables
5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
Misconceptions
Student Misconceptions and Common Errors
- Students need to be given experiences where they make predictions or estimate probabilities, then discover whether or not their intuitions or reasoning were correct.
Vignette
In the Classroom
Matching Dogs to Owners
Summary
The "Matching Dogs to Owners" activity addresses the research question "Are humans able to match dogs to their owners better than blind luck?" To answer this research question, students simulate blind luck with the "just guessing" model. They then compare the results of the "just guessing" model with how well humans match dogs to their owners.
Learning Goals
This activity has the following goals for students:
- Begin to develop understanding of the reasoning process of statistical significance
- Understand that if an observed result is very unlikely under a particular model, then the result provides strong evidence against that particular model
- Engage students in statistical thinking and working as a group
(Students discuss their answers to the questions above in small groups; Note: David Moore is a statistician)
Teacher: Let's share as a large group now for what you think about these questions. Question one just focuses on how many you were able to match correctly. What were the range of correct matches people had?
Student: 0
Student: 3
Student: 2
Teacher: Did anyone match more than 3? Okay so it looks like our range was from 0 to 3 correct. For question 2 what did your groups talk about?
Student: Our group felt that it would be difficult to match up 4 correctly without having some ability to match dogs and owners. We felt there must have been a strategy that he was using.
Student: Our group talked a little about that, but we also felt that it could just be lucky. It might not happen often to get 4 out of 6 correct by guessing, but it is possible.
Teacher: So it seems like the first group was saying that it might not be plausible that he could guess and get 4 out of 6 correct, but the second group mentioned that it is possible. This ties into question 3 then. Let's hear from a couple other groups for this question.
Student: We talked about that both cases could have a case made for them. Four correct matches does give some evidence that it was not just guessing. However, on any given day someone might be able to just get lucky.
Student: Our group talked about wondering if there was a way to tell how likely matching up a certain amount of owners and dogs would be in order to make a better decision on if it was just blind luck or not.
Teacher: Well, that leads into what we will look at next. Read through the following information and then we will discuss how we can simulate the blind luck model of guessing. Both groups had good ideas that we should keep in mind as we work through this part of the lesson.
(Students can work in pairs and use playing cards to perform 50 trials to see how many times 4 or more correct matches happen. Students can use 12 playing cards with 6 pairs of the same card; for example two Ace's, two 2's, two 3's, two 4's, two 5's, and two 6's. Students can deal out the twelve cards in pairs and then count how many of the six pairs were matched. Students should shuffle or mix up the cards after each trial. Data from the trials can be combined to have a larger number of total trials after students have each done 50 trials. Alternatively, the program Tinkerplots can be used to run many trials at once. Students answer the questions below after working.)
1. Based on the results of the simulation, what is the approximate probability of four (or more) correct matches out of 6 under the "just guessing" model David Moore paired dogs with owners completely at random? How strong is the evidence against the "just guessing" model?
2. Explain in your own words, as if to a friend not taking this course, the reasoning process by which we conclude that if someone matches 4 out of 6 dogs to their owners correctly, this result does provide evidence that they have some dog matching ability, beyond random guessing.
Teacher: So let's go ahead and look at your answers to the questions above based on your simulation and also our total class results of the simulation.
Number of Matches |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
Group |
-------- |
--------- |
--------- |
---------- |
--------- |
---------- |
--------- |
1 |
27 (54%) |
22 (44%) |
1(2%) |
0 |
0 |
0 |
0 |
2 |
38 (76%) |
7 (14%) |
5 (10%) |
0 |
0 |
0 |
0 |
3 |
39 (78%) |
10 (20%) |
1 (2%) |
0 |
0 |
0 |
0 |
4 |
35 (70%) |
9 (18%) |
6 (12%) |
0 |
0 |
0 |
0 |
5 |
40 (80%) |
10 (20%) |
0 |
0 |
0 |
0 |
0 |
6 |
29 (58%) |
14 (28%) |
5 (10%) |
2 (4%) |
0 |
0 |
0 |
7 |
30 (60%) |
17 (34%) |
3 (6%) |
0 |
0 |
0 |
0 |
Class Totals |
238 (68%) |
89 (25.4%) |
21 (6%) |
2 (.001%) |
0 |
0 |
0 |
Note: Numbers in parentheses are the percents for each group (50 trials) while the class totals are the percents out of 350 trials.
Student: Not one of our groups had 4 out of 6 matched together or 5 out of 6 or 6 out of 6. It appears unlikely that Moore was just guessing when matching the dogs.
Student: Yes, I would say the evidence is pretty strong, because looking at 50 trials or 350 trials it appears very unlikely that someone could match 4 owners and dogs out of 6 by just guessing.
Teacher: So do you think that it is still possible? Could someone match 4 out of 6 by just guessing?
Student: I think it is possible, because it could happen still. It is something that is realistic, but it just might not happen that often.
Teacher: So you are saying since it is possible then getting 4 out of 6 or more correctly matched has to be larger than 0 percent, but it just might be a small percent.
Student: We might have to just run more trials of the simulation to have it happen.
Teacher: In your groups you talked about your answer to question 2, let's have a group share your answer.
Student: Based on the results of the simulation the probability of getting 4 out of 6 or more correct matches by guessing alone is unlikely. It is possible, but there is strong evidence that there might be something more going on then just guessing.
Student: I would say that in the pictures that were shown there are clues that owners tend to look like their dogs, which could help a person correctly match them.
Teacher: To close class I will have you take a look at the questions below to see if based on your answers to the dogs and owners pictures if there is evidence that you do better than just guessing.
Exit questions.
References:
http://www.tc.umn.edu/~catalst/ (University of Minnesota Introductory Statistics course: Change Agents for Teaching and Learning Statistics (CATALST ) project)
Roy, M.M., & Christenfeld, N.J.S. (2004). Do dogs resemble their owners? Psychological Science, 15, 261-36.
Resources
Teacher Notes
- Students may need support in realizing the different results that maybe attained by running larger amounts of trials for simulations or experiments.
In this lesson, students analyze the fairness of certain games by examining the probabilities of the outcomes. The explorations provide opportunities to predict results, play the games, and calculate probabilities. Students should have had prior experiences with simple probability investigations, including flipping coins, drawing items from a set, and making tree diagrams. They should understand that the probability of an event is the ratio of the number of successful outcomes to the number of possible outcomes. This lesson was adapted from "Activities: Explorations with Chance," which appeared in the April 1992 issue of the Mathematics Teacher.
This lesson plan presents a classic game-show scenario. A student picks one of three doors in the hopes of winning the prize. The host, who knows the door behind which the prize is hidden, opens one of the two remaining doors. When no prize is revealed, the host asks if the student wishes to "stick or switch." Which choice gives you the best chance to win? The approach in this activity runs from guesses to experiments to computer simulations to theoretical models. This lesson was adapted from an article written by J. Michael Shaughnessy and Thomas Dick, which appeared in the April 1991 issue of the Mathematics Teacher.
Web Links for Data Analysis and Probability
Additional Instructional Resources
- Journal of Statistics Education
An international journal on the teaching and learning of statistics
This site provides standards, data sets, lessons and websites at the K-4, 5-8, and 9-12 levels.
- Activities from the Texas Instruments Classroom Activities Exchange can be used to supplement lessons on the concepts in gr. 9-12 Data Analysis and Probability
probability model. A probability model is used to assign probabilities to
outcomes of a chance process by examining the nature of the process. The set
of all outcomes is called the sample space, and their probabilities sum to 1.
simulation a technique used to model probability experiments for real-world applications.
Reflection - Critical Questions regarding the teaching and learning of these benchmarks:
- How can you better plan your lessons in the future so students have experiences with making predictions or estimating probabilities with relevant real world situations?
- How well structured were the activities for students to explain their reasoning or intuitions and then confront or recognize any misconceptions that they had?
- How well can students explain where they will need knowledge of probability in their lives?
- Were students able to explain or demonstrate their knowledge in different representations (language, real world situations, pictorial, symbols, and manipulatives)? How can you facilitate students' translations between representations for more conceptual knowledge?
Materials - suggested articles and books
American Statistical Association. (2007). Publications for assessment and instruction in statistics education.
Haberman, M. (1991). The pedagogy of poverty versus good teaching. Phi Delta Kapan, December, 291-294.
Peck, R., Starnes, D., Kranendank, H., & Morita, J. (2009). Making sense of statistical studies: Teacher's module. Alexandria, VA: American Statistical Association.
This book consists of 15 hands-on investigations that provide students with valuable experience in designing and analyzing statistical studies. It is written for an upper middle-school or high-school audience. Each investigation includes a descriptive overview, prior knowledge that students need, learning objectives, teaching tips, references, possible extensions, and suggested answers.
This is the K-12 portion of the American Statistical Association (ASA) website. They have workshops and online resources for teachers, useful websites, student competitions, and a list of publications in statistics education.
Rosenstien, J., Caldwell, J., & Crown, W. (1996). New Jersey mathematics curriculum framework. Trenton, NJ: New Jersey State Department of Education.
SciMath Minnesota. (1997). Minnesota k-12 mathematics framework. St. Paul, MN: SciMath.
Yates, D. S., Starnes, D S., & Moore, D. S. (2005) Statistics through application. New York: W.H Freeman
Assessment
- (DOK level 3 strategic thinking) Jeremy plays soccer. He scores a goal in 40% of his games. Jeremy wants to design a simulation using a spinner to predict the probability that he will score a goal in 8 out of 10 games. Which simulation design has an appropriate device and a correct trial?
A Divide a spinner into 5 equal sections labeled 1, 2, 3, 4, and 5. Spin the spinner 8 times.
B Divide a spinner into 5 equal sections labeled 1, 2, 3, 4, and 5. Spin the spinner 10 times.
C Divide a spinner into 4 equal sections labeled 1, 2, 3, and 4. Spin the spinner 8 times.
D Divide a spinner into 4 equal sections labeled 1, 2, 3, and 4. Spin the spinner 10 times. (Maryland State Department of Education, 2008, Algebra/data analysis public release, p.7)
Differentiation
- Strategies: Real world problem solving, multiple entry points, vary teaching methods, group work, teach problem solving strategies.
- Challenges: motivation, slower processing, reading and writing ability, organization, and behavior issues and coping strategies.
Teachers must be explicit in how they talk of vocabulary terms and use vocabulary in context.Teachers should use vocabulary terms often so that students will become familiar hearing them in context. Students should also be allowed to practice the use of vocabulary in small groups.
- Strategies: model vocabulary, manipulatives, speak slowly, visuals, variety of assessments, group work, verbalize reasoning, understanding context or concept, making personal dictionaries.
- Challenges: Vocabulary and Reading ability, standardized tests, how to approach problem solving, and cultural differences.
Minnesota Council for the Gifted and Talented
- Strategies: Tiered objectives, open-ended problem solving, grouping (heterogeneous and homogeneous), curriculum compacting, and independent investigations.
- Challenges: Motivation, acceleration and attitude associated with this for students, maturity, isolation and social issues, and not wanting to be moved outside of age group.
Parents/Admin
Administrative/Peer Classroom Observation
Students are... |
Teachers are... |
conducting probability experiments. |
facilitating student learning by structuring activities for students to work in groups. |
creating hypotheses. |
leading discourse that allows students to communicate their ideas. |
investigating relevant real world problems. |
presenting engaging problems that allow student understanding to build on others ideas. |
providing justification for their ideas. |
engaging students by using technology. |
refining their strategies and ideas. |
instruction allows students to develop ideas and construct their knowledge. |
thinking critically about events and probabilities. |
|
"The best way to approach this content is with open-ended investigations that allow the students to arrive at their own conclusions through experimentation and discussion." (New Jersey Mathematics Curriculum Framework, 1996, p. 371).
Parent Resources
- This is an article about the growing variety of jobs in statistics that are available due to the advancements in technology. Lohr, Steve. (2009). For Todays Graduate, Just one word: Statistics. New York Times.
- National Library of virtual manipulatives
- This website has a variety of applets and activities for students to explore patterns and investigate probability.
- High School Statistics Resources for Teachers, Parents, and Students.
- This website has summary information of other websites that can be helpful for further information, practice, and exploration for students.
- Search You Tube for instructional videos on probability and statistics
- List of websites that have probability activities that are in line with the Massachusetts Curriculum Framework Document.
- This website has collections of articles regarding statistics and probability. It contains numerous current events that relate to statistics.