Four activities in this lesson give examples of probability problems with unexpected answers. The
goal of the lesson is to demonstrate that people should be careful when using probability, and
that some games that seem fair are not. The discussion helps users to draw conclusions from the
activities.

Objectives

Upon completion of this lesson, students will:

have seen a variety of activities demonstrating probability

have learned to make observations about the results of the activities

know about conditional probability

have drawn conclusions about the unexpected results of the probability activities

Standards Addressed:

Grade 10

Statistics and Probability

The student demonstrates a conceptual understanding of probability and counting techniques.

Grade 6

Statistics and Probability

The student demonstrates a conceptual understanding of probability and counting techniques.

Grade 7

Statistics and Probability

The student demonstrates a conceptual understanding of probability and counting techniques.

Grade 8

Statistics and Probability

The student demonstrates a conceptual understanding of probability and counting techniques.

Grade 9

Statistics and Probability

The student demonstrates a conceptual understanding of probability and counting techniques.

Number and Quantity

Quantities

Reason quantitatively and use units to solve problems.

Statistics and Probability

Conditional Probability and the Rules of Probability

Understand independence and conditional probability and use them to interpret data

Use the rules of probability to compute probabilities of compound events in a uniform probability model

Making Inferences and Justifying Conclusions

Understand and evaluate random processes underlying statistical experiments

Make inferences and justify conclusions from sample surveys, experiments, and observational studies

Using Probability to Make Decisions

Calculate expected values and use them to solve problems

Use probability to evaluate outcomes of decisions

Advanced Functions and Modeling

Data Analysis and Probability

Competency Goal 1: The learner will analyze data and apply probability concepts to solve problems.

6th Grade

Data Analysis and Probability

The student will demonstrate through the mathematical processes an understanding of the relationships within one population or sample.

7th Grade

Data Analysis and Probability

The student will demonstrate through the mathematical processes an understanding of the relationships between two populations or samples.

8th grade

Data Analysis and Probability

The student will demonstrate through the mathematical processes an understanding of the relationships between two variables within one population or sample.

7th Grade

Probability and Statistics

7.14 The student will investigate and describe the difference between the probability of an event found through simulation versus the theoretical probability of that same event.

Student Prerequisites

Arithmetic: Student must be able to:

use addition to make estimations about the outcomes of experiments

work with simple fractions

Technological: Students must be able to:

perform basic mouse manipulations such as point, click and drag

use a browser for experimenting with the activities

If the
Crazy Choices game is experimented with "by hand" students will need several (at least one for each
student, more is better) different random number-generating devices, for example some of the
following:

dice with various numbers of sides

spinners

bags of numbered lotto chips, or chips of several colors, or marbles of several colors

Remind students what they have learned about probability in previous lessons:

Ask students to recall what probability is.

Ask students to recall the difference between experimental and theoretical probability.
Briefly discuss the Law of Large Numbers.

Objectives

Let the students know what it is they will be doing and learning today. Say something like this:

Today, class, we will be talking more about probability in relation to several different
games. Often the likelihood of winning a game seems to be the same each time the game is
played when in actuality the likelihood of winning it is not the same.

After demonstrating the game, point out to students the game will keep track of the necessary
statistics:

number of games played

number of times each player won

experimental probability of winning

Guided Practice

Students can play the game in groups (2-10 people per group) using computer(s) or various
random number generating devices (dice, spinners, etc.).

If students play the game using hands-on materials, they may want to keep track of this data
using the
Crazy Choices Game Tally Table that can be reproduced for each group of students. Students should play a lot of games
(50-100) if they want to obtain reliable statistics. The goal of the game is to determine
which player has better chances of winning if players use different devices to determine
whether they win. For example, to compare the chances of the player who flips a coin (winning
in 1 out of 2 possible outcomes) and the chances of the player who rolls a six-sided die
(winning if it rolls a 1 or 2, or in 2 out of 6 possible outcomes).

Next, introduce the
Two Colors Game, where students will learn about conditional probability. Groups of students can play the
game many times, first trying to predict or guess their chances of winning, and keeping track
of the results using the
Two Colors Tally Table.

Have the students play Monty Hall. Most students do not expect the answer to the Monty Hall
problem to be as it is. Each student or group of students can try to solve the problem and to
explain the solution. Then they can run the experiment on computers or by hand, comparing
experimental data with their solutions. Groups of students can discuss why their theoretical
answers agree or do not agree with the data.

The
Monty Hall Multiple Trials activity will allow students to see the results of running the Monty Hall applet many times,
thereby obtaining accurate data quickly, and allowing the teacher to explain this problem
without spending a large amount of time collecting data.

Closure

Conclude the lesson with the
Think and Check! discussion to leads students through the solutions to the activities used in this lesson.

Alternate Outline

This lesson can be rearranged in several ways.

If class time is limited, choose only one of the activities and have students use the computer
version only, which will give fast results while demonstrating the concepts of conditional
probability thoroughly.

If more time is available, have the students try out the activities using dice, spinners, red
and green chips, index cards, etc. to understand what the computer is simulating, and how
quickly the trials can be run on the computer.

After these discussions and activities, the students will have seen more problems that explain
what probability is, and be introduced to conditional probability. The next lesson,
Introduction to the Concept of Probability, further explains the concept of probability and the basic set operations that are useful in
solving probability problems that involve counting outcomes.