 statistics    |    fractals    |    linear inequalities    |    graph theory probability & game theory    |    non-euclidean geometry

Probability & Game Theory Lesson Plan

Abstract

This lesson is designed to improve students' understanding of decision making strategies by introducing students to basic probability.

Standards (NCTM)

Data Analysis

Understand and apply basic concepts of probability
• understand and use appropriate terminology to describe complementary and mutually exclusive events
• use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations
Student Prerequisites

• Educational:

Students should know:

• basic concept of data sets/collection.
• basic statistical concepts and measures

• Technological:

Students must be able to:

• perform basic mouse manipulations such as point, click and drag.
• use a browser such as Netscape for experimenting with the activities.
Teacher Preparation

Teachers will need: Students will need:
• pencil and paper.

Lesson Outline

1. Focus and Review
• Have students discuss the chance that a fair coin will land on heads if it's flipped.
• Present the students with the Birthday Problem. Ask them: "What is the probability you will share a birthday with someone in this room?"
• After giving the students time to figure out the problem, walk the class through the solution.
• Inform the class that today they are going to learn about different types of probabilities and how they relate to game theory.
1. Objectives
• Students will be introduced to the concept of proportionality.
1. Guided Practice
• Split the students into groups of two or more.
• Hand out the bags of M&Ms - one for each group.

• Students should:
• Count the M&M's in their bag by color.
• Record the results on a piece of paper.
• Record the total number of M&M on the same piece of paper.
• Have students draw 60 M&M's with replacement and record the results by color.
• After they finish ask them to calculate the proportion of M&M's drawn of each color.
1. Teacher Input
• Discuss how the groups' experimental results could be different from the theoretical probabilities.
• Discuss the fact that knowledge of previous trials does not affect the probability of getting a heads on subsequent trials. Introduce this type of event as an independent event.
• Discuss how the results could be different if the students did not replace the M&M's
Guided Practice

• Split students up into groups of two or more.
• Hand out the Conditional Probability Data Sheet.
• Specify the number of times students will need to repeat the procedure.
• Allow students time to work through the activity.
• Upon finishing, let students further explore this concept by playing with the Marbles applet. Make sure the set up is the same as it was for the Conditional Probability Exercise.
Teacher Input

• Does each combination have the same probability of being drawn? Which combinations have a smaller chance and why?
• Give formal definitions of notation for the probability of an event, the probability of its complement, and its conditional probability - the probability that event A will occur based on the fact that event B has already occured.
Teacher Input

• Refer back to the brain teaser about the common birthdays and explain that the question can be answered with conditional probabilities.
Guided Practice

• Have pairs of students run sets of 1000 trials for groups of different sizes with the Birthday Problem Activity.
• After collecting each groups data, graph the probablity of repeated birthdays as a function of number of people in the group.
Teacher Input

• Explain Bayes Theorem.
• Address how it applies to game theory.
• Run through the Bayes Theorem activity with the class to illustrate this point.
Guided Practice

• Allow students to experiment with the Bayes Theorem activity by entering in different values to observe the effects of changing the various probabilities on the situation.
• Have students record the trends they notice on a piece of paper.
• After explaining how the activity uses Bayes Theorem to predict your next move, allow students to play the Rock-Paper-Scissors game.
• Again students should note any trends they notice while playing the game.
Teacher Input

• Review/Define basic probability notation
• Pose the question of the Prisoner's Dilemma:

• Imagine two criminals arrested under the suspicion of having committed a crime together. However, the police do not have sufficient proof to convict them. The two prisoners are isolated from each other; the police visit each of them and offer a deal: they will receive a reduced sentence if they rat on their partner. If neither criminal accepts the the offer, there is only enough evidence to convict each of them for 5 years. If only one criminal offers evidence, he goes to prison for 1 year and his partner will go to prison for 10 years. However if both criminals take the deal, each of them will get the maximum number of years, because the deals are null and void after enough evidence has been gathered to convict the criminals.

 Prisoner 1 Prisoner 2 Stay Silent Take Deal Stay Silent 5 , 5 10 , 1 Take Deal 1 , 10 10 , 10

• Discuss the idea of game theory as optimization of strategies.
1. Independent Practice
• Allow students to play through the Friends and Fortunes Activity while carefully considering whether to use aggressive or conservative strategies.
• Have students try the game again using different strategies.
1. Closure
• Discuss the results of the Friends and Fortunes game and the different strategies the students used.
• Lead a discussion about the Magnet Schools article and how it applies to game theory.

The Shodor Education Foundation, Inc.