Tree Diagrams Meeting
Topic
provided by The Actuarial Foundation®
www.actuarialfoundation.org
Students will learn how tree diagrams can be used to understand the different possible outcomes of given
events.
Materials Needed
♦♦ Copies of the Branching Out with Tree Diagrams problem set (Problems and answers can be viewed
here. The problem set is available for download from www.mathcounts.org on the MCP Members Only
page of the Club Program section.) This complete activity, as well as three more related activities, can
be downloaded directly from www.actuarialfoundation.org/programs/youth/shake_rattle_roll.shtml. See
the Teaching Guide and Activities link.
Meeting Plan
OBJECTIVES: Students will understand—
• How to us a tree diagram to map outcomes and determine the probability of different events occurring
• Mutually exclusive and complementary events
WHAT YOU WILL DO:
1. Introduce the actuarial profession briefly. Discuss tree diagrams. Explain that tree diagrams can be
used to determine the probability (or chance) of a particular scenario occurring. Explain that property and
casualty actuaries think about different outcomes caused by natural events and costs related to each
outcome. They may advise insurance companies about how the costs might affect future claim payments
amounts.
2. Distribute copies of the Branching Out
with Tree Diagrams activity (shown here) to
students. (If you are using the document from
www.actuarialfoundation.org/programs/youth/
shake_rattle_roll.shtml, this is Reproducible
Activity 4.)
3. Discuss the Saffir-Simpson Hurricane
Scale used to categorize hurricanes
according to strength. The scale is a 1-5
rating based on the hurricane’s present
intensity and can be used to estimate the
potential property damage and flooding
expected from a hurricane landfall. The scale
is based on wind speed and storm surge.
4. Explain to students that for the purposes
of this activity, each scenario mapped out in
each tier of the tree diagram is equally likely.
(This is not always the case in tree diagrams.
Though they show all of the possible
outcomes, each possible outcome is not
always equally likely.) Using the tree diagram,
demonstrate how to calculate the probability
of a possible outcome. Note that the results
in a later tier of the diagram strongly depend
2011–2012 MATHCOUNTS Club Resource Guide
Branching Out with Tree Diagrams
Name: ___________________________________________
Date: _____________________________
Hurricanes are measured based on wind speed and storm surge. The Saffir-Simpson Hurricane Scale
ranks categories of hurricanes from Category 1 (weakest) to Category 5 (strongest). A tropical storm is
weaker than a Category 1, but stronger than a tropical depression. It takes more than eight hours for most
hurricanes to change in intensity.
16 Hours Later
Property and casualty actuaries think
about and list different outcomes caused
by natural events. Then they determine
costs related to each outcome. These costs
may affect premium prices in the future.
The tree diagram to the right shows some
of the possible outcomes of a Category 1
hurricane over time. Study the tree diagram
carefully and then answer the questions. For
the purposes of this exercise, assume each
event represented in each tier is equally
likely.
Cat 3
8 Hours Later
Cat 2
Cat 2
Cat 1
Hurricane
Identified
Cat 1
Cat 2
Cat 1
Cat 1
Tropical Storm
Tropical Storm
Cat 1
Tropical Storm
Tropical Depression
Work the Math:
1. List the possible events that could happen eight hours after a hurricane has been identified.
_____________________________________________________________________________________
2. What is the probability that this hurricane is a Category 1 after 16 hours? Write your answer as a
common fraction and a percentage to the nearest whole number.
__________________________
3. What is the probability that this hurricane remains a Category 1 for the entire time? Write your answer as
a common fraction and a percentage to the nearest whole number.
__________________________
4. What is the probability that this hurricane is a Category 2 after 16 hours? Write your answer as a
common fraction and a percentage to the nearest whole number.
__________________________
5. After 16 hours, is it more likely that the hurricane will be a Category 1 or a Category 3? Explain your
answer using fractions.
_____________________________________________________________________________________
6. If flooding is possible in any of these scenarios, would a storm and a flood be mutually exclusive events?
_____________________________________________________________________________________
37
on the outcome of the prior tier. For example, the outcome of “Tropical Depression” is dependent on the
outcome of “Tropical Storm” in the prior tier.
5. Explain that mutually exclusive events cannot happen at the same time. For example, a Category
1 hurricane and a Category 4 hurricane are mutually exclusive. Complementary events are all the
other outcomes that do not occur in a given scenario. For example, if an event is “it will rain today,” the
complementary event will be “it will not rain today.” If an event occurs, its complement cannot occur.
Answers to questions on Branching Out with Tree Diagrams activity:
1. After 8 hours, a hurricane can stay the same (Category 1), increase intensity (Category 2) or decrease intensity (tropical
storm); 2. 1/3 and 33%; 3. 1/9 and 11%; 4. 2/9 and 22%; 5. A Category 1 is more likely. The probability of a Cat 1 is 1/3,
while the probability of a Cat 3 is 1/9.; 6. No, mutually exclusive events cannot happen together.
Possible Next Steps
Visit www.actuarialfoundation.org/programs/youth/shake_rattle_roll.shtml for the original version of this
activity and three, other related activities (Teaching Guide and Activities). You also will find PDF files for a
poster, additional classroom activities and family activity pages to download and use with your students!
NCTM Standards Met by This Lesson:
Data Analysis and Probability
Understand and apply basic concepts of probability
Number and Operations
Understand numbers, ways of representing numbers, relationships among numbers, and number systems
Compute fluently and make reasonable estimates
Problem Solving
Reasoning and Proof
Communication
Connections
Representation
This meeting plan is taken from the
Shake, Rattle, & Roll Teaching Guide
created by The Actuarial Foundation.
38 2011–2012 MATHCOUNTS Club Resource Guide
Branching Out with Tree Diagrams
Name: ___________________________________________
Date: _____________________________
Hurricanes are measured based on wind speed and storm surge. The Saffir-Simpson Hurricane Scale
ranks categories of hurricanes from Category 1 (weakest) to Category 5 (strongest). A tropical storm is
weaker than a Category 1, but stronger than a tropical depression. It takes more than eight hours for most
hurricanes to change in intensity.
16 Hours Later
Property and casualty actuaries think
about and list different outcomes caused
by natural events. Then they determine
costs related to each outcome. These costs
may affect premium prices in the future.
The tree diagram to the right shows some
of the possible outcomes of a Category 1
hurricane over time. Study the tree diagram
carefully and then answer the questions. For
the purposes of this exercise, assume each
event represented in each tier is equally
likely.
Cat 3
8 Hours Later
Cat 2
Cat 2
Cat 1
Hurricane
Identified
Cat 1
Cat 2
Cat 1
Cat 1
Tropical Storm
Tropical Storm
Cat 1
Tropical Storm
Tropical Depression
Work the Math:
1. List the possible events that could happen eight hours after a hurricane has been identified.
_____________________________________________________________________________________
2. What is the probability that this hurricane is a Category 1 after 16 hours? Write your answer as a
common fraction and a percentage to the nearest whole number.
__________________________
3. What is the probability that this hurricane remains a Category 1 for the entire time? Write your answer as
a common fraction and a percentage to the nearest whole number.
__________________________
4. What is the probability that this hurricane is a Category 2 after 16 hours? Write your answer as a
common fraction and a percentage to the nearest whole number.
__________________________
5. After 16 hours, is it more likely that the hurricane will be a Category 1 or a Category 3? Explain your
answer using fractions.
_____________________________________________________________________________________
6. If flooding is possible in any of these scenarios, would a storm and a flood be mutually exclusive events?
_____________________________________________________________________________________