A Program of The Actuarial Foundation. Aligned with Common Core State and NCTM Standards.
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What is an actuary? An actuary is an expert in statistics who works with businesses, governments, and organizations to help them plan for the future. Actuarial science is the discipline that applies math and statistical methods to assess risk.

Background photo: © Inga Nielsen/iStockphoto.

About This Lesson Plan



Shake, Rattle, & Roll

Branching Out with Tree Diagrams



  • Grade 7: Statistics and Probability (CCSS 7.SP.7 and 8)
  • Grades 6–8: Make Sense of Problems, Model With Mathematics, Use Appropriate Tools Strategically, and Look for and Express Regularity in Repeated Reasoning (CCSS MP1, 4, 5, and 8); NCTM: Data Analysis and Probability
  • Standards Chart: Common Core and NCTM (PDF)


Students will understand:

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


  1. Activity 4 (PDF)
  2. Coin for demonstration
  3. Paper for problems


  1. Branching Out with Tree Diagrams
  2. Answer Key to the Lesson Reproducibles


Introduction to Tree Diagrams

1. Group students into pairs. Show students the coin. Ask students to figure out what the chances are of flipping two heads in a row. Ask students to volunteer how they came up with their answers. Possible solution methods include an organized list, a table, or a tree diagram. Discuss tree diagrams. Explain that they can be used to determine the probability (or chance) of a particular scenario occurring.

2. Guided Practice: Ask students to complete a tree diagram to determine the probability of flipping three heads in a row (1/8 or 12.5%). Discuss answers as a class.

3. Independent Practice: Distribute copies of Branching Out With Tree Diagrams (PDF) for either homework or classwork. To help students understand the data they are going to work with:

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.

  • Check for Understanding: Review answers with the class. Review the following as appropriate for your class: 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.


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

Welcome Families (PDF)
Family Activity 1: Insuring for the Uncertain (PDF)
Family Activity 2: