Conditional Probability and Probability of Simultaneous Events

Abstract

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

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

Links to other standards.

Student Prerequisites

  • Arithmetic: Students 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 such as Netscape for experimenting with the activities

Teacher Preparation

All activities in the lesson are better experienced by using the software, with individual students or small groups of students having enough time to explore the games and find answers to the related questions. If the activities have to be set up physically, the following materials are necessary (one set of materials for each group of students that will be doing the activity):

  • Access to a browser
  • Pencil and Paper
  • Copies of the supplemental materials:
  • For the Racing Game with One Die:
    1. one six-sided die
    2. The Racing Game Table to tally the results
  • For the Two Colors game:
    1. three identical containers (e.g., small boxes or opaque cups)
    2. 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.
    3. The Two Colors Table to tally the results
  • For the Monty Hall activity:
    1. Three identical index cards
    2. The Table to tally the results

Key Terms

This lesson introduces students to the following terms through the included discussions:

Lesson Outline

  1. 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?

  2. 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.

  3. Teacher Input
    • Begin by describing the Racing Game with One Die activity, which shows experimentally how the number of steps in the Racing Game affects the probability of winning.
  4. Guided Practice

    • 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:

      1. The experimental probability of winning the game is not the same as the probability of taking one step. Why?
      2. What would happen to the probabilities if there were more than two steps to the finish?
    • 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:

      1. The experimental probability of winning the game is not the same as the probability of taking one step. Why?
      2. 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.

  5. 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.

  6. 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 Outlines

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.
  • Combine this lesson with the Unexpected Answers lesson.
  • Or choose fewer of the activities to cover; for example, use only the Racing Game with One Die and the Conditional Probability discussion and make the focus conditional probability only. Use the Probability of Simultaneous Events discussion somewhere else in the Probability unit.
  • 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.