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Unit 8, Lesson 9

Using Tables for Conditional Probability

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Integrated Math 2

9.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Math Talk MLR8: Discussion Supports

Materials

None

Activity Narrative

This Math Talk focuses on dividing fractions. It encourages students to think about methods for dividing fractions and to rely on strategies such as multiplying by an inverse to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students compute probabilities by dividing other probabilities in fractional form.

To divide fractions, students need to look for and make use of structure (MP7).

Launch

Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:

Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies, and record and display their responses for all to see.
Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.

Keep all previous problems and work displayed throughout the talk.

Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Activity

None

Find the value of each expression mentally.

Activity Synthesis

To involve more students in the conversation, consider asking:

“Who can restate ’s reasoning in a different way?”
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”

MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. Examples:

“First, I because .”
“I noticed , so I .”

Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing

9.2

Activity

Instructional Routines

Extend It MLR8: Discussion Supports

Materials

None

Activity Narrative

In this activity, students use two-way tables to represent the outcomes in a sample space. Turning the two-way table into a relative frequency table shows estimates for probabilities, which can then be used to decide if events are independent.

Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Give students 10 minutes to work on the problems, and then pause for a brief whole-class discussion.

Action and Expression: Provide Access for Physical Action. Support effective and efficient use of tools and assistive technologies. To use spreadsheet software, some students may benefit from a checklist or a list of steps to be able to use the technology.
Supports accessibility for: Organization, Memory, Attention

9.3

Activity

Instructional Routines

None

Materials

None

Activity Narrative

In this activity, students use a two-way table to estimate probabilities, including conditional and compound events.

Launch

Arrange students in groups of 2. Give students quiet time to work on the questions, have partners compare answers, and then have a whole-class discussion.

Representation: Access for Perception. Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems, and other text-based content.
Supports accessibility for: Language

Activity

None

Lesson Synthesis

Display the table from the Cool-down and its description.

A large aquarium has 48 guppy fish in it. The table summarizes the fish based on color and sex.

	orange	blue
male	5	13
female	12	18

Here are some questions for discussion.

“What is the probability that a fish picked at random is orange?” ()
“What is the probability that a fish picked at random is orange under the condition that the fish is female? ( or equivalent)
“What is the probability that a fish picked at random is orange and male?” ()
“What is the probability that a fish picked at random is orange or male?” ( or equivalent)
“What is the probability that a female fish picked at random is blue?” ( or equivalent)
“How would you write ‘what is the probability that a female fish picked at random is blue?’ using probability notation?”()
“Are the events of being an orange fish and being a female fish independent events?” (They are not independent events. and . If the events were independent, these probabilities would be equal.)

to access Cool-down.

Student Lesson Summary

Organizing data in tables is a useful way to see information and compute probabilities. Consider the data collected from all students in grades 11 and 12 and their intentions of going to prom.

	going to prom	not going to prom	total
grade 11	43	84	127
grade 12	81	35	116
total	124	119	243

A student is randomly selected from this group of students. We can find a number of probabilities based on the table.

because there are 81 students who are both in 12th grade and going to prom out of all 243 students in the group who could be selected.

because there are 116 grade 12 students out of the entire group.

because there are 124 students going to prom out of the entire group.

since there are 159 students in grade 12 or going to prom (43 from grade 11 going to prom, 81 from grade 12 going to prom, and 35 from grade 12 not going to prom).

represents the probability that the chosen student is going to prom under the condition that the student is in grade 12. The group we are considering is different for this probability because the condition is that the student is in grade 12. Imagine all grade 12 students are in a room and we are selecting from only this group. We see that 81 students from this group are going to prom out of the 116 students in the group.

represents the probability that the chosen student is in grade 11 under the condition that the student is going to prom. In this case, we imagine that everyone going to prom is gathered in a room, and we are selecting one student from this group. The probability that this student is in grade 11 when considering only this group is .

The last two probabilities can also be found using the multiplication rule.

can be rewritten .

For example, substituting values into gives which is a true equation.

A pharmaceutical company is testing a new medicine for a disease, using 115 test subjects. Some of the test subjects are given the new medicine and others are given a placebo. The results of their tests are summarized in the table.

	no more symptoms	symptoms persist	total
given medicine	31	26	57
given placebo	16	42	58
total	47	68	115

Divide the value in each cell by the total number of test subjects to find each probability to two decimal places. Some of the values have been completed for you.

no more symptoms symptoms persist total

given medicine 0.27 0.50

given placebo

total 1

If one of these test subjects is selected at random, find each probability:
From the original table, divide each cell by the total for the row to find the probabilities with row conditions. Some of the values have been completed for you.

no more symptoms symptoms persist total

given medicine 0.54

given placebo 1
Jada didn’t read the instructions for the previous problem well and used the table she created on the first problem to divide each cell by the probability total for each row. For example, in the top left cell she calculated . Complete the table using Jada’s method.

no more symptoms symptoms persist total

given medicine

given placebo

What do you notice about this table?
From the original table, divide each cell by the total for the column to find the probabilities with column conditions. Some of the values have been completed for you.

no more symptoms symptoms persist

given medicine 0.66

given placebo

total 1
Are the events “symptoms persist” and “given medicine” independent events? Explain or show your reasoning.
Based on your work, does being given this medicine have an affect on whether symptoms persist or not?

	no more symptoms	symptoms persist	total
given medicine	0.54
given placebo			1

Activity Synthesis

The purpose of this discussion is for students to interpret probabilities in a two-way table.

Ask students to interpret each of the probabilities they found. For example, “What does the probability mean in the context of the problem?”

Here are sample interpretations of the probabilities.

: The probability that a test subject chosen at random from all of the subjects still had symptoms after the test.
: The probability that a test subject chosen at random from all of the subjects was among those given medicine and still had symptoms after the test.
: The probability that a test subject chosen at random from all of the subjects still had symptoms after the test or was given a placebo.
: The probability that a test subject chosen at random from those who were given medicine had symptoms after the test.
: The probability that a test subject chosen at random from those who were given placebo had no more symptoms after the test
: The probability that a test subject chosen at random from those who had symptoms after the test was given medicine.
: The probability that a test subject chosen at random from those who had no more symptoms after the test was given placebo.

Display what Jada did when creating the top left cell in table:

Here are some questions for discussion.

“What do you notice about what Jada did and ?” (I noticed that they are equal.)
“What does the equality look like when applied to general events A and B.” ()
"What is the relationship between this equation and the multiplication rule?" (From the multiplication rule, we know that . If we divide both sides by , we get the equation that Jada used.)
“How did you determine whether the events were dependent or independent?” (I determined that they were dependent events because and were not equal. If the events were independent, these probabilities would be equal.)

MLR8 Discussion Supports. For each observation that is shared, invite students to turn to a partner and restate what they heard, using precise mathematical language.
Advances: Listening, Speaking

A blood bank in a region has some information about the blood types of people in its community. Blood is grouped into types O, A, B, and AB. Each blood type either has the Rh factor (Rh+) or not (Rh-). If a person is randomly selected from the community, the probability of that person having each blood type and Rh factor combination is shown in the table.

	O	A	B	AB	total
Rh+	0.374	0.357	0.085	0.034
Rh-	0.066	0.063	0.015	0.006
total					1

What does the 0.085 in the table represent?
Use the table or create additional tables to find the probabilities, then describe the meaning of the event.

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Lesson 9

Using Tables for Conditional Probability

Warm-up

Standards Alignment

Building On

7.NS.A.2

Addressing

Building Toward

HSS-CP.A.4

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Activity Synthesis

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Lesson Synthesis

Student Lesson Summary

Student Response

Building on Student Thinking

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSS-CP.A.4

Building Toward

Standards Alignment

Building On

Addressing

HSS-CP.A.4

Building Toward