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The area model

Probability, grade 9 Lesson 9 of 11Learning

The area model

The area model is one more way to describe a two-stage experiment, after the outcome table and the tree diagram. In it, a probability is drawn as an area:

  • a square whose side is 1, so its area is 1 – all the experiment’s outcomes together;
  • each outcome of the experiment is a rectangle inside the square;
  • an outcome’s probability is the area of its rectangle.
Building an area model
Why a product: in 35\displaystyle \frac{3}{5} of the draws the first marble is red, and in 25\displaystyle \frac{2}{5} of those the second is blue. 25\displaystyle \frac{2}{5} of 35\displaystyle \frac{3}{5} is 35⋅25\displaystyle \frac{3}{5}\cdot\frac{2}{5}.

The example below builds the area model of a two-stage experiment.
A bag holds 3 red marbles and 2 blue ones.
A marble is drawn, put back in the bag, and another marble is drawn.

  • The first stage: cut the square into upright strips, a strip for each outcome. A strip’s width is the outcome’s probability: 35\displaystyle \frac{3}{5} of the width for red, 25\displaystyle \frac{2}{5} for blue.
  • The second stage: cut each strip across. Each part’s height is the probability of the outcome in the second stage.
  • A rectangle’s area: its width times its height – the product of the probabilities, as along a path in a tree. For example, red and then blue: 35⋅25=625\displaystyle {\frac{3}{5}\cdot\frac{2}{5}}={\frac{6}{25}}.

The marble goes back into the bag, so the second stage does not depend on the first: the cut is at the same height in both strips.

One experiment as a tree and as an area model

A tree diagram and an area model show the same numbers:

  • a branch in the first stage is a strip, and the probability on it is the strip’s width;
  • a branch in the second stage is a part of a strip, and the probability on it is the part’s height;
  • a path is a rectangle, and the path’s probability is the rectangle’s area.

The example below describes one experiment both ways.
A bag holds 3 red marbles and 2 blue ones.
A marble is drawn, not put back, and another marble is drawn.

Without putting back, the second stage depends on the first, so the two strips are cut at different heights:

  • after a red, 2 of the 4 marbles left are red – half the height;
  • after a blue, 3 of the 4 marbles left are red – three quarters of the height.
An event in the area model
The rest of the square is the complement, “two different colours”. Its probability: 1−25=35\displaystyle {1-\frac{2}{5}}={\frac{3}{5}}.

An event is several rectangles, so an event’s probability is the sum of its rectangles’ areas.

The example below finds the probability of the event A\displaystyle A – “both marbles are the same colour”, for the bag of the example before (without putting back). The event’s rectangles are marked: red and red, blue and blue.

P(A)=620+220=820=25 P(A)=\frac{6}{20}+\frac{2}{20}=\frac{8}{20}=\frac{2}{5}
Exercise 1

A spinner has three parts: half of it is red, a quarter blue and a quarter yellow.
The arrow is spun twice.

a

Write each rectangle’s area, and mark the rectangles of the event “the same colour twice”.

b

Find the probability of getting the same colour twice.

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    c

    What is the probability of getting two different colours?

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      Exercise 2

      A coin is tossed, and then a spinner is spun.
      A third of the spinner is red, and two thirds of it are blue.

      a

      Write the probabilities in the tree diagram.

      b

      Write the areas in the area model.

      c

      Which number in the tree equals the area of the rectangle “tails, blue”?
      What are this rectangle’s width and height, and where are they in the tree?

      Exercise 3

      A bag holds 2 green marbles and 3 yellow ones.
      A marble is drawn, not put back, and another marble is drawn.

      a

      Write each rectangle’s area, and mark the rectangles of the event “at least one green marble”.

      b

      Find the event’s probability: add the areas you marked.

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        c

        Find the same probability with the complement.

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          Exercise 4

          Here is the area model of an experiment: a marble is drawn from a bag, and then another one.

          a

          How many marbles were in the bag at the start, and how many of each colour?

          All the marbles

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            Red

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

              Blue

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                b

                Was the first marble put back before the second draw? Explain from the model.

                c

                Find the probability that both marbles are the same colour.

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power