Title
Who's the Smartest (Jo Ann Ellerbrock)
Goals
(1) Students will explore the concept of conditional probability
(2) Students will examine events to determine if they are independent or dependent.
(3) Students will organize probabilities with a tree diagram.
(4) Students will be able to appropriately use the multiplication principle
Abstract
Problem Statement
Sometimes the probability of an event may change if we are told the occurrence of another event. For example, the probability of a person selected at random having lung cancer is not real high. However, if the person is a heavy smoker, then the probability would be higher. This activity will allow students to explore this concept of conditional probability and learn techniques for determining if two events are dependent or independent.
Instructor Suggestions
(1) Discuss with students the difference between independent and dependent events.
(2) Use Venn diagrams to illustrate the need to subtract out overlapping occurrences. (e.g., The number of people taking shop is 20, the number of people taking drafting is 45, and the number of people taking both is 10. What is the probability of choosing someone at random who is either taking shop of drafting?)
(3) Discuss the notation and verbal syntax for conditional probabilities.
(4) Discuss the ways to examine events A and B for independence, i.e., P(A|B) = P(A), P(B|A) = P(B), or P(AB) = P(A)*P(B).
(5) Have students explore the multiplication principle for independent and dependent events.
(6) Have students use a tree diagram to represent the multiplication principles.
(7) The problem is politically correct and indicates that IQ is not dependent relative to sex.
Materials
"Who's the Smartest" Activity sheet, pencil, paper, calculators.
Time
This activity should take one class period.
Mathematics Concepts
Discrete Mathematics Concepts
Counting principles, independent and dependent events, probability, multiplying probabilities.
Related Mathematics Concepts
percents, problem solving,
NCTM Standards Addressed
Problem Solving, Reasoning, Connections, Probability, Discrete Mathematics
Colorado Model Content Standards Addressed
Data Collection and Analysis (3), Problem Solving Techniques (5)
Curriculum Integration
Computationally, this activity is appropriate for any students who has skills in computing percents and basic probabilities. However, the level of logic required makes it more appropriate for students in Algebra II or above. It integrates nicely into any unit on probability.
Further Investigation
Variations/Comments
References/Resources
Crisler, N., Fisher, P., & Froelich, G. (1994). Discrete mathematics through applications. New York: W.H. Freeman and Company
Kenney, M.J., & Hirsch, C.R. (Eds.). (1991). Discrete mathematics across the curriculum, K-12. Reston, VA: National Council of Teachers of Mathematics.
Steen, Lynn A. (Ed.). (1991). For All Practical Purposes. New York: W.H. Freeman and Company