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Why do I multiply
when working with
Why Multiply?
probability?
Connecting the Algorithm
to the Models
This activity builds conceptual understanding of why we multiply when
finding the probability of two independent events using multiple
representations. By examining tree diagrams, organized lists and the
area model, students make conceptual connections from the use of
models to the algorithm for finding the probability of two independent
events both occurring.
Written by Jennifer Brackney and Linda Horst
(During the BITL IV Grant as funded by the Office of the Commissioner of Higher Education in Montana)
Abstract: This activity builds conceptual understanding of why we multiply when finding the
probability of two independent events using multiple representations.
Grade: 8
Strand: Probability
Required Class Time: 1 period
Materials: Student Worksheet, Whiteboard, Whiteboard Markers
Objectives:
1. Establish a productive math disposition.
BITL 4: Probability
Jennifer Brackney & Linda Horst
June 2008
Page 1
2.
3.
4.
5.
Encourage adaptive reasoning.
Multiple representations of probability situations
Find theoretical probability of simultaneous events
Emphasize correct mathematical vocabulary
Vocabulary:
Theoretical probability
Event
Dependent Event
Multiple Events
Favorable Outcome
Area Model
P(A then B) = P(A) •P(B)
Organized List
Outcomes
Independent Event
Tree Diagram
Procedure: For all teacher notes, the provided examples will be based on the following
probability situation.
If you are at a carnival and to win you must spin a one (outcomes: 1, 2, 3, 4)
and you must draw a blue marble (outcomes: blue, red, yellow).
1. Organized List:
One, Blue
Two, Blue
Three, Blue
Four, Blue
One, Red
Two, Red
Three, Red
Four, Red
One, Yellow
Two, Yellow
Three, Yellow
Four, Yellow
As a class, review the process of using an organized list to calculate the probability of
two independent events both occurring. In this discussion make sure to include why we
multiply the probability of event A by the probability of event B. Example: When we
make an organized list we multiply the probability of event A by the probability of event
B because one-fourth of the sample space results in an outcome of one, and one-third of
the sample space results in the outcome of blue. Therefore, we multiply the probability
of event A by the probability of event B because we want to find one-fourth OF onethird. Remind the students that generally, in math the word OF tells us that we want to
multiply.
2. Tree Diagram:
One
Blue
Yellow
BITL 4: Probability
Jennifer Brackney & Linda Horst
Two
Blue
Red
Red
Red
Red
Three
Blue
Yellow
Yellow
Four
Blue
Yellow
June 2008
Page 2
Review as a class the process of using a tree diagram to calculate the probability of two
independent events both occurring. In this discussion make sure to include why we multiply
the probability of event A by the probability of event B. Example: When we use a tree
diagram to figure probability, we multiply the probability of event A by the probability of
event B because we want to find one-third OF one-fourth.
3. Area Model:
Blue
Yellow
1
One
Two
Red
3
¼
Three
Four
BITL 4: Probability
Jennifer Brackney & Linda Horst
June 2008
Page 3
Introduce the process of using the area model to find the probability for situations. Using a
matrix, list all possible outcomes for event A on the vertical axis, and all possible outcomes
for event B on the horizontal axis. Next, shade the regions representing the favorable
outcome(s). Thinking about area and the fractional part of a whole that each outcome
represents on each axis, calculate the area of the shaded region. Again, emphasize the
connection between the model and the algorithm.
4. Student Activity:
Hand out the student worksheet and have students work within their small groups to discuss
and analyze the given examples. Students must connect each of the methods to the
algorithm. Once students finish the student worksheet, each small group of students need to
create their own probability situation, represent the situation using each model and link it to
the algorithm.
BITL 4: Probability
Jennifer Brackney & Linda Horst
June 2008
Page 4