9.4.3 C Applying Probability Concepts
Standard 9.4.3 Essential Understandings
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
188.8.131.52 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.
184.108.40.206 Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes.
220.127.116.11 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.
18.104.22.168 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.
22.214.171.124 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).
126.96.36.199 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.
188.8.131.52 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.
184.108.40.206 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.
220.127.116.11 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 C
18.104.22.168 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).
22.214.171.124 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.
126.96.36.199 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.
What students should know and be able to do [at a mastery level] related to these benchmarks
- Students should be able to determine if two events are dependent or independent
- apply probability concepts such as intersections, unions, and complements of events with different representations including Venn diagrams to solve problems
- Students should be given multiple experiences to develop strategies for working with probability involving intersections, unions, and complements of events that can lead to the understanding of formulas.
Work from previous grades that supports this new learning includes:
- Students should be familiar with using the 0-1 scale as a measure of probability.
- Students have begun to develop intuition about probabilistic situations.
- Students can model real world situations involving both equally and unequally likely outcomes
- Students have had experience with relative frequency
- Students have a variety of techniques for listing outcomes
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.
Student Misconceptions and Common Errors
- interchange the use of the words "and" and "or"
- Venn diagrams are useful tools in organizing. Although disjoint (mutually exclusive events) can be seen using two non-overlapping circles, independence cannot be shown on a Venn diagram.
- Students often confuse independent and disjoint (mutually exclusive) probabilities
- "Research has shown that intuitions about probability are often incorrect, can lead us to incorrect conclusions about chance events, and persist despite age and education. Many of us have heard someone who has had a lengthy run of poor card hands dealt to them say they are certain they will get a good hand soon-they are "due." Ignoring the independence of each uncertain deal of the cards is called "gambler's fallacy" (Minnesota Math Frameworks, 1997, Randomness and Uncertainty, P.4)
In the Classroom
A teacher asked her students if they own a dog or a cat. There were 12 boys and 16 girls in the class. Here are the results:
The following is a lesson based on these data, teachers should gather data from their class.
Teacher: What is the chance that a boy owns a dog?
Student: Well, since there are 12 boys and 10 of them own a dog, the chance of a boy owning a dog is 10 out of 12 or about 83%
Teacher: Good Job. Now, what is the chance that a boy owns a cat?
Student: Easy, that is 3 out of 12 or 25%.
Teacher: OK, what is the chance that a boy owns either a dog or a cat?
Student: Since 83% of boys own a dog and 25% own a cat, then 108% of boys must own either a dog or a cat. Wait a minute that can't be right.
Student: You forgot the boys that own both. Since there are 12 boys and 2 do not own either a dog or cat then there must be 10 that own either a dog or a cat or both. So the chance of a boy owning a dog or a cat is 83%.
Teacher: That's right. All of the boys that own cats also own dogs, so the percentage does not change.
Teacher: What about the girls?
Student: Since 8 out the 16 girls own a dog the chance of a girl owning a dog is 50%.
Student: Yeah, and the chance of a girl owning a cat is 6 out of 16 or 37.5%.
Teacher: What about owning both?
Student: You can't trick us this time, since there are only 4 girls that do not own a dog or a dog, there must be 12 left.
Student: So, 12 out of 16 is 75%. That means that 75% of the girls own either a dog, a cat or both a dog and a cat.
Teacher: Now I am going to place the names of the students into a hat. What is the chance that the first name I pull out of the hat owns a dog?
Student: Well since 10 boys and 8 girls own dogs and since there are 28 students in the class, the chance that the first name you pull owns a dog is 18 out of 28 or about 64%.
Teacher: Good Job, now would your answer change if I told you the name that I pulled out was a girl?
Student: Of course it would, since the percentage of girls that own a dog is not the same as the percentage of boys that own a dog, the answer would definitely change.
Teacher: What if I told you that the name I drew owned a cat. Would you be able to guess the chance that the name drawn is a girl?
Student: Since you told us that the name drawn owns a cat and that there are only 9 people that own a cat, we know that the name drawn must be one of those nine students.
Student: So if the student drawn is one of those the chance that the student picked is a girl is 6 out of 9 or about 67%.
Teacher: To conclude I what to know if not owning a dog or a cat and the gender of a student are independent events.
Student: Independent events are events that do not effect one another, right?
Teacher: That is one way of looking at it. Another way to see is if events are independent is to ask yourself "if I know one of the outcomes does that change the likelihood of the other outcome?" If the answer is yes then then the events are not independent.
Student: Well 12 out the 28 students are boys, but only 2 out of the 6 students that do not own a pet are boys. The first percentage is about 43% and the second one is only 33%.
Student: That means that the chance a randomly chosen student is a boy and the chance that a randomly chosen student does not own a pet are different. That means that the events are NOT independent.
Teacher: Good Job! Tomorrow we can explore more relationships within the data.
- Students may need support in further development of the specific use of vocabulary related to independence, intersections, unions, and complements.
- Teachers need to be diligent on use of the words "and" and "or". Often students use the word "and" to represent the union instead of "or". The use of the word "or" in mathematics in general and probability and statistics specifically, is different than the everyday use of the word. In mathematics "or" means in one group (or event) or in another group (or event) or in both, This is not true in the everyday world. If a waiter asks if you would like "soup or salad", you are expected to choose just one of the options and not both. But in mathematics if you ask for a number that is either even or prime, the number two (2) is in both sets.
Lesson was adapted from "Activities: Explorations with Chance," Mathematics Teacher (April, 1992). 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.
Stick or Switch? Lesson was adapted from an article by J.M. Shaughnessy & T. Dick, Mathematics Teacher (April 1991). 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.
Additional Instructional Resources
- Journal of Statistics Education
This international journal is on the teaching and learning of statistics.
- The Exploring Data-Math Forum
This site provides standards, data sets, lessons and websites at the K-4, 5-8, and 9-12 levels.
- The Texas Instruments Classroom Activities Exchange
The site can be used to supplement lessons on the concepts in Gr. 9-12 Data Analysis and Probability.
conditional probability the probability of an event given some other event.
independently combined probability models. Two probability models are said to be combined independently if the probability of each ordered pair in the combined model equals the product of the original probabilities of the two individual outcomes in the ordered pair.
independent events Two events A and B are independent if the chance that they both happen simultaneously is the product of the chances that each occurs individually; i.e., if P(A and B) = P(A)P(B). This is essentially equivalent to saying that learning that one event occurs does not give any information about whether the other event occurred too.
intersection The intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.
mutually exclusive events that have no common outcomes (the intersection of two events is empty).
simulation a technique used to model probability experiments for real-world applications.
union The union of a collection of sets is the set of all distinct elements in the collection.
Professional Learning Communities
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
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
- (DOK Level 1: Recall) A warning system installation consists of two independent alarms having probabilities of 0.95 and 0.90, respectively, of operating in an emergency. Find the probability that at least one alarm operates in an emergency. (Adapted from TIMSS gr. 12, L-10) (California Mathematics Framwork, Appendix D, 2005, p. 330).
- (DOK Level 2: Basic Reasoning) Arlene and her friend want to buy tickets to an upcoming concert, but tickets are difficult to obtain. Each ticket outlet will have its own lottery so that everyone who is in line at a particular outlet to buy tickets when they go on sale has an equal chance of purchasing them. Arlene goes to a ticket outlet where she estimates that her chance of being able to buy tickets is 1/2. Her friend goes to another outlet, where Arlene thinks that her chance of being able to buy tickets is 1/3.
1. What is the probability that both Arlene and her friend are able to buy tickets?
2. What is the probability that neither Arlene nor her friend is able to buy tickets?
3. What is the probability that at least one of the two friends is able to buy tickets? (CERT 1997, 37)
(California Mathematics Framework, Appendix D, 2005, p. 330).
- (DOK level 3: Strategic Thinking) I roll two standard fair dice and look at the numbers showing on the top sides of the two dice. Let A be the event that the sum of the two numbers showing is greater than 5. Let B be the event that neither die is showing a 1 or a 6. Are events A and B independent? (California Mathematics Framework, Appendix D, 2005, p. 331).
- (DOK level 2: Basic Reasoning): Tara plays a game using 2 bags of game pieces. One bag has 6 blue game pieces and 6 red game pieces. The other bag has ten game pieces numbered 1 through 10. On her turn, Tara must draw one game piece from each bag. What is the probability that she draws a red game piece and an even-numbered game piece?
A) ½ (B) ¼ ( C) 1/30 (D) 1/60 (Colorado Department of Education, 2001 CSAP grade 10 released items, p.14 )
● (DOK level 2: Basic Reasoning):
- (DOK level 1: Recall)A fish tank at a pet store has 3 red fish, 6 blue fish, 7 yellow fish, and 4 orange fish. If one fish is chosen at random from the tank, what is the probability that the fish will be blue or yellow? (Maryland State Department of Education, 2009, Algebra/data analysis public release, p.34)
- 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.
English Language Learners
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: modeling vocabulary, using manipulatives, speaking slowly, using visuals, using a variety of assessments, assigning group work, verbalizing reasoning, understanding context or concept, making personal dictionaries.
- Challenges: Vocabulary and Reading ability, standardized tests, how to approach problem solving, and cultural differences.
- 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.
Administrative/Peer Classroom Observation
conducting probability experiments.
facilitating student learning by structuring activities for students to work in groups.
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.
instructing to allow 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)
- Lohr, Steve. (2009). For Today's Graduate, Just one word: Statistics. New York Times. This is an article about the growing variety of jobs in statistics that are available due to the advancements in technology.
- 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 or Video Google 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.
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