# Branching Out with Tree Diagrams

**OBJECTIVE**

Students will understand:

- How to use a tree diagram to map outcomes and determine the probability of different events occurring.
- Mutually exclusive and complementary events.

Alignment with NCTM Standards (PDF)

**MATERIALS**

Activity 4 (PDF)

**REPRODUCIBLES**

**DIRECTIONS**

- Discuss tree diagrams. Explain that they 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 payment amounts.
- Distribute copies of Branching Out with Tree Diagrams (PDF).
- 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.
- Demonstrate how to calculate the probability of a possible outcome using tree diagrams. This probability can be expressed as a fraction or a percentage. The determination of probability (which would be true in any tree diagram) for any given tier is to count the number of outcomes under consideration in that tier and divide it by the total number of outcomes in that tier. The results in the second tier of this activity strongly depend on the outcome of the first tier. For example, the outcome of "Tropical Depression" in the second tier is dependent on the outcome of "Tropical Storm" in the first tier.
- 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.

Real-World Connections:

Actuaries must consider all of the possible outcomes for a particular event, such as a hurricane, or the likelihood that a storm will occur and spread. They can then take these possibilities a step further and consider the effects of different scenarios on different variables, such as the type of building materials used in structures and whether they can withstand hurricanes.

Background:

For the Saffir-Simpson Hurricane Scale, visit www.nhc.noaa.gov/aboutsshs.shtml.

**HOME CONNECTION**

Welcome Families (PDF)

Family Activity 1: Insuring for the Uncertain (PDF)

Family Activity 2: Banking on Inflation (PDF)

Family Activity 3: Planning for the Future (PDF)