A Program of The Actuarial Foundation. Aligned with Common Core State and NCTM Standards.
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.

SUBJECT
Math, Probability

6-8

DURATION
30 Mins

COLLECTION
The Power of Probability

Lesson 3: Compound Probability

Students use multiple coin flips to calculate compound probability and use the formula for compound probability to calculate the probability of multiple independent events.

OBJECTIVE

• Students will use a tree diagram to derive the formula for compound probability; and
• Students will use the formula for compound probability to calculate the probability of multiple independent events.

MATERIALS

DIRECTIONS
Getting Started

1.    Show the coin to the class and ask what is the probability of the coin landing on heads.

2.    Recall the work done in Lesson 1 on tree diagrams and ask for a volunteer to explain how all the outcomes for three flips in a row could be depicted.

3.    Ask what the probability of a coin landing on heads three times in a row would be. Walk the students through a tree diagram to demonstrate that there is one outcome for three heads in a row while there are seven unfavorable outcomes (HHT, HTT, HTH, TTT, TTH, THT, THH), so the probability is one out of eight or 1/8 or 12.5%.

4.    Ask if there is a way to calculate the probability of tossing three heads in a row without setting up the tree diagram. If necessary, point out that the probability of one heads flip is 1/2 or 50% or .5 and, for three heads in a row:
1/2 x 1/2 x 1/2 = 1/8 or
.5 x .5 x .5 = .125 or
50% (.5) x 50% (.5) x 50% (.5) = 12.5% or .125
If the class is familiar with exponents, demonstrate that the probability of any number of heads in a row equals 1/2 (number of flips).

5.    On the board, write the formula for compound probability, i.e., the probability of two independent events: P(AB) = P(A)•P(B). Explain the formula and that A and B are independent events: the probability of A occurring does not affect the probability of B occurring, and vice versa.

6.    Provide an example or two to show how the formula works. For example, if the probability of rain is 20% on Saturday and 50% on Sunday, what is the probability of rain on both days? (20% times 50% = 10%). Demonstrate how this could be done with fractions (2/10 times 1/2 = 2/20 which reduces to 1/10) or decimals (.2 times .5 = .10). To reinforce the abstract concept of compound probability, use a tree diagram to demonstrate how rain on both days represents one out of 10 possible outcomes.

7.    Distribute Worksheet 3. Read the introduction and review the facts with the class. Ask students to complete the worksheet. Review answers as a class. Worksheet Answer Key (PDF). As a bonus activity, have students complete the Bonus Worksheet.

LESSON EXTENSION
Have students use the Online Probability Challenge to practice using probability skills for real-life purposes. This interactive online activity challenges students to use probability to help Rick and Athena plan a summer concert tour. This activity can be used as an in-class lesson activity or an out-of-the classroom extension.

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