What “condition” are you in?
(A review of conditional probability)
Authors: Bethany Hadley, Sharon Flinspach, Stacy Schultz, and Deidra Baker
Overview of Lesson: Given two-way frequency tables from various sources, some already
filled in, others with gaps, students will answer various questions about everyday situations
using conditional probability.
Common Core State Standards Grade Level Content
S-CP. 5 - Recognize and explain the concepts of conditional probability and independence in
everyday language and everyday situations.
Common Core State Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
Prerequisites
Students will need to be familiar with sample spaces, two-way frequency tables, marginal and
joint probability, and independent events. This lesson is intended to be a summative activity
covering conditional probability.
Learning Objective
After an exploration activity and a class discussion, students will be able to understand, fill in,
and use two-way frequency tables to apply and explain compound events from everyday life.
Time Required
This lesson will take place in one (to two) 50-minute lesson(s).
Materials Required
Paper, pencil, worksheets
Instructional Lesson Plan
I. Essential Questions or Big Mathematical Ideas
1) How is conditional probability used to interpret 2 way frequency tables?
2) Is P(A|B) always the same as P(B|A)? Why or why not?
3) How can the addition rule be used to fill in blanks on a two-way table?
II. Launch
[Teacher notes: Start the lesson with the questions/discussion:
“Suppose you are the counselor at school, and students have given you the classes they
want to take next year. You are in charge of making the master schedule (explain this
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This lesson plan was developed during an Iowa Department of Education funded Mathematics and Partnership grant in 20132014 titled, "Statistics and Probability for Teaching Secondary Students".
idea). What are some of the problems or conflicts you might have trying to do this?”
“What do you think the probability is of a student being in both chemistry and algebra 2?”
(For example, use classes that are relevant to your students)
Hopefully, the idea that students in chemistry are usually juniors, so are lots of kids in
algebra 2 and English 11, so all those classes can’t be at the same time, especially if
there is only section of each. Students can then discuss where conflicts occur, and why
students may not be able to take the classes they want, when they want them.
Start with the discussion questions in the teacher notes, and then have students look at
the Venn diagram. Given a Venn diagram, students will make a two way data table, as a
quick review of Venn diagrams and how they are related to two-way tables. This is the
first page of the attached worksheet. This can be displayed on the board to class, or
given to students as a single page handout. Students will answer these questions
individually then share in pairs or small groups. Conduct a brief class discussion of
responses to check that all students have the prerequisite knowledge to begin the
lesson. (Venn diagram on worksheet page 1)
III. Plan
Students will be given the body sheets of the worksheet (pages 2 - 4). Students will
work in pairs to answer questions on worksheet. Teacher will monitor group discussions
and ask questions to guide as needed. In question 2, students may have the same
solution for a. and b. and also for questions d. and e. If this is the case, talk to the
student(s) and address the “given condition” in each question. Students may also need a
reminder of the notation for conditional probability. A teacher prompt for the class, or a
check point to verify students are using the notation correctly is appropriate after
question a. The whole class could also do question a. together, and then have students
work in pairs. Question 2g is a checkpoint/class discussion question.
As all groups complete the questions following the second table, have a short
discussion/checkpoint to answer questions or dispel misconceptions. Question 2g is one
of the key ideas for the lesson. Be sure that all students can answer this question (rarely
are these the same!)
Students can then continue answering questions. Another checkpoint/discussion should
take place as most students have table number 3 filled in, maybe having different groups
fill in parts of a class table. As students finish question 4a, have pairs check with other
each to see if they agree and discuss differences. Expect discussion about “young
adults” and “older adults” and who is included in each age group. The intent of this
question is to highlight that ambiguous words are frequently used in advertising.
Students can discuss what these words mean to each person or group. Have a class
discussion as needed about this. Finish the worksheet.
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This lesson plan was developed during an Iowa Department of Education funded Mathematics and Partnership grant in 20132014 titled, "Statistics and Probability for Teaching Secondary Students".
IV. Close
Discuss answers to question 5. Summarize why P(A|B) ≠ P(B|A) using complex
fractions as well as joint and marginal probability ideas from the tables. Write a class
formula for this rule: P(B|A) = P(A and B)/ P(A). Discuss strategies for filling out the
table when there is overlap of categories, and when the people in each category are
separate- the addition rule, this should be review.
Assessment
Included assessment can be used as an exit slip, if time permits, or given as homework.
Possible Extensions
Have students write their own problems based on data they are interested in, such as
make and color of cars, or number and type of pets. The students will need to provide
some data (they can “invent” the data or search online) and then write 3 or 4 conditional
probability questions. Challenge students to write problems that have 0 or 1 as the
possible solution, and have them explain how they know the solution will be 0 or 1.
What conditions need to be met for this to be the case? If there is not a lot of time for
early finishers, they can just answer the last questions.
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This lesson plan was developed during an Iowa Department of Education funded Mathematics and Partnership grant in 20132014 titled, "Statistics and Probability for Teaching Secondary Students".
What “condition” are you in? Launch
There are 54 students in the sophomore class. The Venn diagram below shows how many of
them take Geometry class; participate in soccer, or do both.
______________
18
12
8
Geometry Class
Soccer Club
1. How many students are taking Geometry?
2. How many students play soccer?
3. How many students do both?
4. How many students do not play soccer and do not take Geometry?
5. How many students play soccer but do not take Geometry?
Use the data above to complete the frequency table below.
Takes Geometry
Does Not Take
Geometry
Total
Plays Soccer
Does Not Play
Soccer
Total
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This lesson plan was developed during an Iowa Department of Education funded Mathematics and Partnership grant in 20132014 titled, "Statistics and Probability for Teaching Secondary Students".
What “condition” are you in?
Name: ______________________
1. There are 150 children at summer camp and 71 signed up for swimming. There were a total
of 62 children that signed up for crafts and 28 of them also signed up for swimming.
Use the data above to complete the frequency table.
Takes Swimming
Classes
Does Not Take
Swimming Classes
Total
Makes Crafts
Does Not Make
Crafts
Total
a. How many campers did not take swim class or make crafts?
b. What percentage of the campers who did not take swimming lessons or make crafts?
c. Are campers more likely to take swimming lessons or make crafts? Explain your reasoning.
d. Create a Venn diagram from this two-way table.
2. The table below represents data collected from a freshman survey of how 9th graders get to
school most often. Complete the two-way table:
Male
Female
Total
Walk
Car
28
45
Bus
12
27
Bike
17
69
Total
129
92
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This lesson plan was developed during an Iowa Department of Education funded Mathematics and Partnership grant in 20132014 titled, "Statistics and Probability for Teaching Secondary Students".
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This lesson plan was developed during an Iowa Department of Education funded Mathematics and Partnership grant in 20132014 titled, "Statistics and Probability for Teaching Secondary Students".
a. Suppose:
(A) = student is female
(B) = she walks to school
What is the probability a student walks to school, given the student is female? Rewrite
the question using conditional probability notation and write the probability as a fraction.
b. Suppose:
(A) = student is female
(B) = she walks to school
What is the probability a student is female, given the student walks to school? Rewrite
the question using conditional probability notation and write the probability as a fraction.
c. Are the fractions you wrote for questions a. and b. the same?
d. Suppose:
(A) = student is male
(B) = he rides the bus
What is the probability a student rides the bus, given the student is male? Rewrite the
question using conditional probability notation and write the probability as a fraction.
e. Suppose:
(A) = student is male
(B) = he rides the bus
What is the probability a student is male, given the student rides the bus to school?
Rewrite the question using conditional probability notation and write the probability as a
fraction.
f. Are the fractions you wrote for questions a. and b. the same?
g. From the above examples, you may conclude that the probabilities of event “A” given
condition “B” and event “B” given condition “A” are not the same, P(A|B) ≠ P(B|A). Explain
why this is the case and how you choose the numerator and denominator in this general case.
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This lesson plan was developed during an Iowa Department of Education funded Mathematics and Partnership grant in 20132014 titled, "Statistics and Probability for Teaching Secondary Students".
3. Heather, a hairdresser, is making a record of all the customers she had in the last month.
a. Provide row and column headings for the table below so it can show the number of male
and female customers who are adults and children.
Total
Total
b. In the last month she had 40 female children and only 5 male children.
Put these values in the table.
c. How many total children were customers last month?
d. She has 100 total customers and a total of 20 male customers during the same month.
Use this information to finish filling in your table.
e. Which of the cells represent a marginal probability?
f. Which of the cells represent a joint probability?
4. A large group of people was surveyed about their favorite movie genre. The participants gave
their age and chose their favorite movie genre, or type of movie, from Action, Comedy, and
Horror.
Action
Comedy
Horror
Total
18 - 25 years old
238
450
312
1,000
26 - 49 years old
350
472
178
1,000
50+ years old
320
490
190
1,000
Total
908
1,412
680
3,000
a. Find P(comedy | 26 - 49).
b. Find P(50+ | Horror)
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This lesson plan was developed during an Iowa Department of Education funded Mathematics and Partnership grant in 20132014 titled, "Statistics and Probability for Teaching Secondary Students".
c. Find P(Horror | 50+)
d. A company that sells a product designed for young adults wants to advertise at a
movie. Which genre should they choose? Explain your reasoning.
e. If the company wants to advertise to older adults, which genre would be least
beneficial to advertise in?
5. Using the information in the launch question, write 3 conditional probability statements that
will have a large impact on the master schedule, for the guidance counselor to consider as
the master schedule is built.
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This lesson plan was developed during an Iowa Department of Education funded Mathematics and Partnership grant in 20132014 titled, "Statistics and Probability for Teaching Secondary Students".
What “condition” are you in? Assessment
Name____________________________
The two-way table below shows the number of houses of various styles in a large town.
Exterior
material
Brick
House
Color
Brown
Stucco
Aluminum
Siding
Wood
shakes
12
3
1
2
9
21
1
1
9
2
7
Blue
2
White
2
5
Red
1
3
Total
12
28
Total
1) What is P(brown house| aluminum siding), in fraction form?
2) What is P(aluminum siding| brown house), in fraction form?
3) Are your answers the same? Using the information in the table and your fractions, explain
your answer.
4) Explain your answer to (3) using another representation such as a picture, Venn diagram, or
tree diagram.
5) What is the probability that a house is brown? Is this joint or marginal probability?
6) What is the probability of a house is red and made of stucco? Is this joint or marginal
probability?
10/10
This lesson plan was developed during an Iowa Department of Education funded Mathematics and Partnership grant in 20132014 titled, "Statistics and Probability for Teaching Secondary Students".