Skip to content Skip to navigation

Connexions

You are here: Home » Content » Portraits of Probabilities

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

    • External bookmarks
  • E-mail the author

Recently Viewed

This feature requires Javascript to be enabled.

Portraits of Probabilities

Module by: Interactive Mathematics Program

Intent

Students use area models to represent the outcomes of several probability situations. They also create situations to fit given area models.

Mathematics

This activity continues to reinforce the connection between area models (in the form of rug diagrams) and probabilistic situations.

Progression

Have students do this activity individually, likely for homework, following Rug Games. In class, students share ideas in order to build upon the variety of examples of probabilistic situations and associated area models.

Approximate Time

5 minutes for introduction

25 minutes for activity (at home or in class)

15 minutes for discussion

Classroom Organization

Individuals, followed by whole-class discussion

Doing the Activity

Part II of this activity is the first time students are given a rug diagram and are asked to describe a situation to match it. You may want to brainstorm a few possibilities for the rug diagram in Question 6. For example, the diagram might represent the probability that the tire that went flat on your new car was the front left tire.

Discussing and Debriefing the Activity

Part I: Rugs for Events

You might want to give transparencies and pens to five groups and have each group prepare a visual aid to lead the discussion about that situation. Each group may elect to synthesize its members’ answers or use one particular member’s answer.

As presentations are made, ask the rest of the class, What is another way to draw a rug for this situation?

For instance, Question 1 asks students to draw a rug with a shaded portion that represents a probability of 1616 size 12{ { {1} over {6} } } {} . Students will likely divide a rug into six equal sections, with one shaded. This could be done in various ways, such as those shown below.

Figure 1
Figure 1 (graphics1.jpg)

Students could also draw a rug with just two sections, one representing 1616 size 12{ { {1} over {6} } } {}, as illustrated below. and the other representing 5656 size 12{ { {5} over {6} } } {}, as illustrated below.

Figure 2
Figure 2 (graphics2.jpg)

Students will probably explain Questions 2 and 3 using a theoretical model. Questions 4 and 5 bring up the extreme cases of a probability of 0 or 1.

Note that Question 3 is similar to Item D in What Are the Chances? in which students were asked to find the probability of flipping a coin twice and getting different results. Some students may still see the flipping of a coin twice as having three equally likely possible outcomes—two heads, one head and one tail, and two tails. It is important that they realize that these three outcomes are not equally likely.

In What Are the Chances? a list was suggested as a means of analyzing the two-coin situation. Now you might ask students how the two-coin situation can be represented with a rug diagram.

    Coin 1
    H T
Coin 2 H Both heads Coin 1: TCoin 2: H
  T Coin 1: HCoin 2: T Both tails

Drawing a rug diagram with four equal parts doesn’t in itself explain why the four sequences are equally likely. The fact that the columns are of equal size represents an assumption that coin 1 is fair. The equal-size rows represent the fairness of coin 2. These fairness assumptions include the assumption that the tosses of coins 1 and 2 are independent events.

Displaying such a diagram may spark some insight and help to convince some students that the three outcomes—two heads, one head and one tail, and two tails—are not equally likely.

Using two different coins (such as a penny and a dime) can help students recognize that each rectangle represents a distinct outcome. Ask, If you flip heads on the dime, what might happen with the penny? How would you show that in a rug diagram?

Part II: Events for Rugs

As students share the situations they created, expect a variety of responses, including that some of the rug diagrams stumped some students.

Key Questions

What is another way to draw a rug for this situation?

If you flip heads on the dime, what might happen with the penny? How would you show that in a rug diagram?

Supplemental Activity

Heads or Tails? (extension) expands on the two-coin situation by asking students to investigate the probabilities for the outcomes of flipping three, four, and ten coins.

Comments, questions, feedback, criticisms?

Send feedback