This lesson is based on several interesting problems. Each problem has a somewhat unexpected
answer; in fact, many people have a hard time accepting experimental results for these problems,
as the results may seem counterintuitive. This very difference in expectations and actual results
leads to a deeper consideration of the related mathematics and to acquiring new tools for solving
problems, namely the ideas and formulas connected with conditional probability and probability of
simultaneous events.

Objectives

Upon completion of this lesson, students will:

have taken a closer look at conditional probability

have learned the formula for probability of simultaneous independent events

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.

Seventh Grade

Statistics and Probability

Investigate chance processes and develop, use, and evaluate probability models.

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

Grades 9-12

Data Analysis and Probability

Understand and apply basic concepts of probability

Advanced Functions and Modeling

Data Analysis and Probability

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

Discrete Mathematics

Data Analysis and Probability

Competency Goal 2: 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

7.15 The student will identify and describe the number of possible arrangements of several objects, using a tree diagram or the Fundamental (Basic) Counting Principle.

Student Prerequisites

Arithmetic: Student must be able to:

use addition, subtraction, multiplication and division to solve probability formulas

understand how tables can be used in multiplication

Technological: Students must be able to:

perform basic mouse manipulations such as point, click and drag

use a browser for experimenting with the activities

Teacher Preparation

Access to a browser

Pencil and paper

Copies of supplemental materials for the activities:

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. (The objects have to fit in the containers and have to be indistinguishable
from each other by touch.)

The chances of something happening, based on repeated testing and observing results. It is the ratio of the number of times an event occurred to the number of times tested. For example, to find the experimental probability of winning a game, one must play the game many times, then divide the number of games won by the total number of games played

probability

The measure of how likely it is for an event to occur. The probability of an event is always a number between zero and 100%. The meaning (interpretation) of probability is the subject of theories of probability. However, any rule for assigning probabilities to events has to satisfy the axioms of probability

random number generators

A device used to produce a selection of numbers in a fair manner, in no particular order and with no favor being given to any numbers. Examples include dice, spinners, coins, and computer programs designed to randomly pick numbers

theoretical probability

The chances of events happening as determined by calculating results that would occur under ideal circumstances. For example, the theoretical probability of rolling a 4 on a four-sided die is 1/4 or 25%, because there is one chance in four to roll a 4, and under ideal circumstances one out of every four rolls would be a 4. Contrast with experimental probability

Lesson Outline

Focus and Review

Remind students what has been learned in previous lessons that will be pertinent to this lesson
and/or have them begin to think about the words and ideas of this lesson

If I bet you that we could play a game and that I could win every time, would you believe me?

This game is a racing game in which we take turns rolling a six sided die and advancing on the
numbers that we each are assigned. I bet you I can assign us an equal quantity of numbers that
we move on and no matter how many times we play I will always win.

Then tell them that the numbers that you assign yourself are 1, 2, 3, 4, 5, and 6, while the
numbers you assign the person who takes you up on your bet are 7, 8, 9, 10, 11, and 12. (If
you are only playing with one die then it is impossible to roll anything higher than a 6 so
the person assigned 6 -12 will never move.)

Who thinks this game is fair?

Objectives

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

Today, class, we are going to begin learning about probability

We are going to use the computers to learn about probability, but please do not turn your
computers on until I ask you to. I want to show you a little about this activity first.

Have groups of students play the
Racing Game with One Die either using the software (preferably) or rolling a six-sided die and using the
Table to tally the results.

Players in the game should have unequal chances to take a step. Knowing the probability of
each player taking a step, students can try to predict the probability of each player winning
the game, and try multiple experiments in order to test the prediction.

Lead a discussion about the
Probability of Simultaneous Events to introduce the formula for probability of simultaneous independent events.

This discussion is based on the results of the
Racing Game with One Die. Each group of students can think about and discuss the following questions, later discussing
them with other groups and with their mentor:

The experimental probability of winning the game is not the same as the probability of
taking one step. Why?

What would happen to the probabilities if there were more than two steps to the finish?

Next, initiate a discussion based on
Conditional Probability. This discussion requires the active participation of the mentor. If there are students who
want to take on the role of mentors, they can read the discussion ahead of time in order to
prepare. This way discussions can happen in smaller groups.

Independent Practice

Have the students use the
Two Colors game to perform experiments that will demonstrate conditional probability.

There are three closed boxes. One box contains two green balls, another one contains two red
balls and the last one has one red and one green ball. If students use the software, the
computer will shuffle the boxes. If students use manipulatives, one of them should shuffle the
boxes. A student chooses one box and picks one ball from it (without looking). If the first
ball is red, the game starts over. If the first ball is green, the student wins if the second
ball in the same box is also green.

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
Table.

Closure

You may wish to bring the class back together for a discussion of the findings. Once the
students have been allowed to share what they found, summarize the results of the lesson.

Alternate Outline

This lesson can be rearranged in several ways.

Include the
Monty Hall problem to further clarify conditional probability. Each student or group of students can try
to solve the problem and explain the solution. Then they can run the experiments on computers
or by hand (in the latter case, recording the results in the
Table), comparing experimental data with their solutions. Groups of students can discuss why their
theoretical answers agree or do not agree with the data.

Use the
Think and Check! discussion to help students understand the explanation of the Monty Hall problem and the Two
Colors Game.

Have students come up with their own version of the
Two Colors game, and present their game and probability results to the class.

Suggested Follow-Up

After these discussions and activities, the students will have worked with condition probability
and have seen the formula for the probability of simultaneous events. The next lesson,
From Probability to Combinatorics and Number Theory , is devoted to data structures and their applications to probability theory. Tables and trees
are introduced, and some of their properties are discussed