Applications of Probability Instructional Notes
Based on CCSS Mathematics II: Unit 4
Common Core Academy: Day 3
Participants should have completed the reading assignment: 5 Practices for Orchestrating Productive Mathematics Discussions, Chapter 4.
Goals:
Create and understand appropriate representations of probability outcomes, including diagrams and graphs. Use probability to evaluate outcomes and make decisions.
Understand independence and conditional probabilities and use these understandings to interpret data.
Probability Standards Addressed in These Tasks:
S.CP.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”).
S.CP.4Construct 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.
S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
S.CP.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.
S.CP.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.
Learning Cycle: Representations of Categorical Data & Conditional Probability
Goal: Define and organize sample spaces using 2-way tables, Venn diagrams, and tree diagrams.
The objectives are:
1.Define and organize sample spaces in a variety of ways.
2.Use 2-way tables, Venn diagrams and tree diagrams to show and interpret characteristics of outcomes, including: “and”, “or”, “not”, “complement”, “conditional”.
3.Given one type of diagram or table, be able to generate the other representations and make connections between representations.
4.Use probability to recognize characteristics about sample spaces and make decisions.
5.Use the language and symbols of probability appropriately.
Introduction to the Probability Unit
Display posters throughout all tasks: Learning Cycles & Mathematical Practices
Materials needed:
Exploring Data.ppt: PowerPoint display of a two-way table
Before leading the task, be sure you have previewed the PowerPoint.
Task #1: This activity models an activity designed by Steven Leinwand (author of Accessible Mathematics). The mathematical goal of this activity is to make sense of the data.
Launch: Show the PowerPoint that displays a partially filled out table. Each subsequent slide adds a bit more data to the table. Show one slide at a time, letting participants guess what might belong in the missing cells, and what the data might represent.
Explore: Accept all possible responses. Continue to add information to the table bit by bit as participants refine and explain their conjectures until the final piece of information is revealed.
Discuss:
1.Before the column and row labels are added, emphasize the importance of context.
What is in the missing cells?
What might the data represent?
What can be discovered from the table?
What do you need to know in order to understand the table?
Listen for ideas about the necessity of unit labels, and table percentages. Without context the data tends to be meaningless. Context is important – gives relevancy to data! Share participants’ ideas and share work that brings out upcoming topics on table percentages, units of measure, and comparisons between categories.
2.What did you find you NEEDED to know in order to understand the table? (units of measure, column and row labels, etc.)
3.Reflect on the Practice Standards that were employed in the activity (1, 3, 6 & 7), specifically attending to precision in mathematical language.
4.Introduce the idea of how mathematical language may differ from common usage. The Mean Teacher comic illustrates this and is at the end of PowerPoint. (This issue should be revisited throughout the day, for example in the discussion of “and” and “or” in probability.)
Develop Understanding Task
Task #2:TB or Not TB?
Surfaces ideas such as:
●a review of 2-way tables and marginal (row and column) totals along with joint frequencies (interior cells of the table) which have already been encountered in Secondary Math 1.
●proportions may be more useful than counts when comparing data, but counts give actual values and the size of sample spaces.
●using 2-way tables to calculate probabilities/rates/proportions from table values.
●conditional probabilities reveal information about subgroups that may not be immediately apparent from casual examination of the table.
●using the results of the exploration to promote the need for probability notation.
●using data and probabilities to make recommendations and decisions.
Materials needed: “TB or Not TB?” task sheets, calculators, a method for displaying student responses such as giant post-it posters or document cameras
Launch: Discuss the prevalence of tuberculosis (TB) in countries with inadequate health care. Infection rates in some countries are as high as 30%, and consequently affects people traveling from country to country and is deadly to many individuals, especially those with compromised immune systems. All food handlers are required to take the TB skin test, which involves injecting a bubble of serum underneath the skin, then several days later, observing the skin to see if the person has shown a reaction to the serum, which indicates the presence of TB antibodies.
Review information about tuberculosis:
Famous people with tuberculosis:
Explore: Distribute page 1 of the task sheet. If the pages are stapled, separate the papers. Allow time for participants to think about how they would approach the question “Do I really have tuberculosis?” As participants explore the data table they will discover that high numbers of TB skin tests result in a false positive result.
After a sufficient time for exploration, distribute page 2 of the task, which provides guided questions.
Discuss:
1.How does exploring conditional probabilities allow for investigation of the accuracy of medical tests?
Students may have difficulty with identifying the wording of conditional probabilities, for instance: “out of” and “given”.
As results are shared to the class, drive the need for labeling their results with correct “units”, which will demonstrate the efficiency of probability notation.
On a giant sticky write, using correct notation, the answers to questions #1-4 on “Tuberculosis or Not, Part II”.
For example: “The probability of having TB given that an individual tests positive”
= P(TB | positive) = 361/423 = 0.853 = 85.3%
2.How do we define “accuracy” in context of these medical tests?
3.Discuss number sense, and what the chance of a false positive really means. Is the probability of a false positive a high chance or a low chance? How low is low? Compare to lottery probabilities.
4.In this setting a false negative is much worse than a false positive. In what settings might a false positive be worse? How would you feel if you got a false positive?
5.The rates are based on large population values. What do you tell an individual who asks “Do I have TB?”
Solidify Understanding/Develop Understanding Task: (Two-way tables to Venn diagrams)
Task #3:Grandma’s Birthday
Surfaces ideas such as:
●exploring the relationship between 2-way tables and Venn diagrams
●accurately representing a sample space, and accounting for each outcome
●relating spaces in a Venn diagram to the language of probability, including “and”, “or”, “complement” and “conditional”
Materials needed: “Grandma’s Birthday” task sheet, calculators
Launch: Grandma Addams is a character from both old TV series, movies and cartoons called “The Addams Family”. She was a crazy old lady whose favorite hobby is wrestling alligators. No one is entirely certain whose mother Grandma is. When discussing Grandma’s living arrangements, Morticia said to Gomez “Your mother came to visit and, apparently, never left.” In surprise Gomez replied: "I thought she was your mother!" It doesn’t matter who she belongs to, our friends and relatives are going to give her a birthday party.
Explore: Distribute the task sheet and allow groups time to create a Venn diagram. There are at least 4 different Venn diagrams that can be created from the data given, dependent on which categories are chosen to be represented by the circles of the diagram. The lack of labels on the diagram is deliberate. Most will choose Friend and Relative as the labels for the circles. As the Venn diagrams are created, look for different representations.
Common errors in creating Venn diagrams relate to today’s instructional theme “How are students thinking about mathematics”. Common errors may include:
1.Labeling the two circles Related & Not Related, then not knowing what to do about the category of Friend, and vice versa.
2.Not subtracting the overlapping amount from both Friend and Relative. (Model norms of a math classroom by showing what can be learned from common errors.)
3.The total number in the party room may not equal 200.
4.When drawing Venn diagrams from scratch, students may draw the circles but forget to account for the outcomes that are not in the circles.
5.When answering the question P(Friend or Relative) students will not know if the two sets are inclusive. In normal English usage, “or” is exclusive. In probability notation, “or” is inclusive.
Participants should finish the task sheet and the questions establishing the connections between 2-way tables and Venn diagrams.
Discuss:
1.Start with displaying a participant’s representation that has inaccurate totals in the Venn diagram.
2.Where does Grandma belong? Are “Friend” and “Relative” disjoint/mutually exclusive (no overlap, no data belongs to both sets)?
3.Point to an area outside of the circles and ask “Who does this represent?”, then ask for proper notation. Repeat for other areas in the diagram.
Mention that there should be no attempt to draw Venn diagrams to scale. These diagrams display relationships between the data, but not the size of the data sets.
4.In making the connection between Venn diagrams and the two-way tables, make note that there are 4 areas in the Venn diagram and there are 4 matching interior cells on the two-way tables. Ask participants to sketch a Venn diagram directly on top of their 2-way table.
5.Explore the questions at the bottom of the worksheet. One thing that can be seen from a two-way table is the totals. Venn diagrams help clarify relationships. Participants may have other points to bring out.
6.Discuss the vocabulary of marginal totals and joint frequencies.
7.Discuss abbreviations for probability areas, and notation including “and’, “or”, and “complement”. Ask how participants can engage prior learning by relating probability notation to other set notations, including solutions to absolute value inequalities ( |x| < 5; x<5 and x>-5).
8.What is the probability that a randomly selected person from the party is either a friend or a relative = P(Friend or Relative)?
Talk about using counts vs. percents and the advantages of each.
Find the probability of P(Friend or Relative), using the two-way table.
Generalize: illustrate P(A or B) by shading the Venn Diagram
9.Develop the equation of the addition rule: P(A or B) = P(A) + P(B) - P(A and B).
Support document: Sally’s Error
Mrs. Bollinger instructed her students to make a Venn Diagram representing the equation:
P(A or B) = P(A) + P(B) - P(A and B)
Sally made this diagram. Is she correct or incorrect? What was Sally thinking about as she made her diagram? Explain. (What was Sally’s error?)
10. Disjoint/Mutually Exclusive: build a 2-way table that displays disjoint sets, and think of a context that can be represented by your table.
What does a two-way table look like for disjoint sets? (Some cells would be 0). What is the universal set for your data? What is your population?
Ask: P(A) + P(B) - P(A and B) = ????
This is the addition rule for probability. Now apply the addition rule to disjoint sets.
11.Is it important to discuss common errors with students? Why?
STOP! Pedagogical content knowledge in mathematics teaching
Regarding the tasks Exploring Data, TB or Not TB, and Grandma Addams’ Birthday
- Review the content standards that apply to the tasks, S.CP.1 and S.CP.7.
- Review the Mathematical Practices that were integrated into the tasks.
- Ask participants where these tasks fit within the learning cycle.
- What about these tasks provided a low threshold?
- What about these tasks provided a high ceiling?
- What were the cognitive levels of these tasks?
- Discuss the 5 Practices for Orchestrating Productive Mathematics Discussions. Refer to Chapter 4.
- Where did you see the 5 Practices in the previous tasks?
Develop Understanding/Solidify Understanding/Practice Understanding Task:
(Develop the idea of independence. Solidify and practice 2-way Tables, Venn Diagrams and conditional probabilities.)
Surfaces ideas such as:
●finding conditional probabilities and using those probabilities to make conjectures about independence between categories
●recognizing the features of a two-way table indicates that a relationship is independent (proportionality of a category to the total is maintained for conditions within the population)
●explaining independence relationship in context of events
●understanding that variability and sample sizes from random sampling might impact probability decisions
●giving evidence to support conclusions
Task #4: Titanic Tragedy
Materials needed: task sheets, calculators, markers & giant sticky notes, OR document camera (Note: if the pages are stapled together, remove the staple and hand out pages separately.)
Review information about the Titanic
Launch: Ask: What do you know about the Titanic tragedy? (Janet: “Well, it sank.”)
Provide additional background about the Titanic tragedy.
Explore: Distribute the “Titanic Tragedy Part I” & “Titanic Tragedy Part II” task sheets and allow participants to explore the data. Choose some individuals or groups to share their graphical representations, numerical calculations, and conclusions about conditional probabilities.
●We can expect to see Venn diagrams and 2-way tables because of the prior tasks. Participants may also choose to compare data using bar graphs or circle graphs. Point out that a bar graph does not show the relationship between survival and gender.
●Compare and contrast different representations and ask participants to find similarities and parallels. Look for representations that differ because counts were used rather than percentages in comparing categories.
●Segmented bar graphs can lead to rich discussions about representing data, what kinds of information can be obtained by displaying in counts vs. percents, and how the graphs change.
●This task uses categorical data. It is important to define displays of data carefully so that the diagrams don’t have cases where an individual both dies and survives, or is both male and female. This quality makes it difficult when thinking about how to construct a Venn diagram, or how to label columns and rows in 2-way tables.
●There are two variables represented in this task. If you are creating a two way table, one variable represents the rows of data, and one represents the columns.
●The idea of showing independence between categories is difficult to grasp and typically takes exposure to multiple data sets, including both examples and non-examples, to gain a solid understanding. It is important to bring out the idea that survival from the Titanic Tragedy was dependent on various categories.
●Look for evidence of understanding that if survival rate is dependent on gender (not independent) then the percent of surviving women will be different than surviving men, and in fact the percent of surviving women or men will be different than the overall survival rate.
●Ask participants “What is the probability of survival?” Discuss how you decide how to determine that probability and record on sticky the different probabilities of survival, P(Survive), P(Survive|Female), P(Survive|Male) and compare. (If time allows other probabilities could be compared.)
●On the top half of a poster write:
P(Survive) =
P(Survive|Male) =
P(Survive|Female) =
Leave the bottom half of the poster to fill in after “Titanic Tragedy III”.
Titanic Tragedy Part II. This task asks participants to check the the data for independence for survival rate vs. class of passenger. It can be used as desired.
Titanic Tragedy Part III:
Distribute Part III
●Participants should fill out the table with numbers that they have determined would indicate that gender has no effect on survival rate. They may struggle to find values to fill in the table that show independence between gender and survival rate, but encourage them to create values that are reasonable.
●Note that there are thousands of ways for the data to show dependence, but only one way that shows independence (with a small bit of rounding). The correct solution is when the overall survival rate is equal to the survival rates of both men and women.
●Participants may try to guess and check or may show proportional arguments. If participants are stumped, prompt them to use the overall survival rate.
Discuss:
1.Ask participants to summarize the characteristics of data that would indicate independence between categories, ie, that proportionality between categories of subgroups will equal each other, and that of the overall population.
2.What would a segmented bar graph showing proportions look like for this data? (The bars would be of equal length, and the “survive” portions would be equal at about 30%)
3.Formalize the proportionality relationship of independent events with the probability equation, P(A)=P(A|B) [i.e., P(Survive) = P(Survive|Female)=P(Survive|Male)]
4.Fill in the bottom half of the poster, adding a heading “In an alternate universe...”
P(Survive) =
P(Survive|Male) =
P(Survive|Female)=
Why are these probabilities equal?
5.Have participants formalize a definition of independence both in a sentence and using probability notation. Write these on a poster for display.