Unexpected Answers
Abstract
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
The activities and discussions in this lesson address the following
NCTM standards:
Data Analysis and Probability
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;
- compute probabilities for simple compound events, using such methods as
organized lists, tree diagrams, and area models.
Student Prerequisites
- Arithmetic: Students 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 such as Netscape for experimenting with
the activities
Teacher Preparation
Students will need:
- Access to a browser
- Pencil and Paper
- Copies of the following supplemental materials:
- 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
- coins
- For doing the Two Colors Game
by hand, students will need:
- three identical containers (e.g., small boxes or opaque cups)
- six objects of two different colors (three of each color), such as marbles or poker chips
- Two Colors Tally Table to tally the results.
The objects have to fit in the containers and have to be indistinguishable from each other by touch.
- For playing the Single Trials
Monty Hall Game without computers, students will need
- three identical index cards
- Monty Hall Tally Table to tally the results.
Lesson Outline
- Focus and Review
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, sometimes games seem like the
likelyhood of winning is the same each time the game is played when in
actuality it is not.
- Teacher Input
- Show the students how to play the Crazy Choices
Game.
- 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.
- Describe the Monty Hall
Problem, based on the familiar game show.
- Independent Practice
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 Outlines
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.
- Combine this lesson with Conditional probability and
probability of simultaneous events, which deals with conditional
probability in more depth.
Suggested Extensions
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.
|