Unexpected Answers

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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.


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:

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

Teacher Preparation

  • 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:
    1. dice with various numbers of sides
    2. spinners
    3. bags of numbered lotto chips, or chips of several colors, or marbles of several colors
    4. coins
  • For doing the Two Colors Game by hand, students will need:
    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
    3. 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
    1. three identical index cards
    2. Monty Hall Tally Table to tally the results.

Lesson Outline

  1. 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.

  2. 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.

  3. 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:
      1. number of games played
      2. number of times each player won
      3. experimental probability of winning

  4. 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.

  5. 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.

  6. 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.
  • Combine this lesson with Conditional probability and probability of simultaneous events, which deals with conditional probability in more depth.

Suggested Follow-Up

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.

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