Chapter 7 of 8

All 8 chapters
  1. 1Collecting data, readunderstood
  2. 2Bar charts and pie charts, readunderstood
  3. 3Mean, median and mode, readunderstood
  4. 4Comparing two groups, readunderstood
  5. 5Line graphs and honest charts, readunderstood
  6. 6The language of chance, readunderstood
  7. 7Working out chances, readunderstood
  8. 8Experiments and fair games, readunderstood

M3.7.7 Chapter 7

Working out chances

A dice has six faces. How do I work out the chance of an even number?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Find a probability when all outcomes are equally likely
  • Find the chance that something does not happen

Each possible result is an outcome. Rolling a dice has six outcomes: 1, 2, 3, 4, 5 and 6. If a fair dice is rolled, the outcomes are equally likely. None of them is favoured.

Draw it first

The chance of an even number

  1. Step 1: the six faces of a dice, numbered 1 to 6. Step 2: the faces 2, 4 and 6 are hatched. Step 3: a label says chance 3/6 = 1/2.123456

    1 List all the outcomes of the dice: 1, 2, 3, 4, 5 and 6. There are 6.

  2. Step 1: the six faces of a dice, numbered 1 to 6. Step 2: the faces 2, 4 and 6 are hatched. Step 3: a label says chance 3/6 = 1/2.123456

    2 Mark the outcomes that count. Here they are the even numbers: 2, 4 and 6. There are 3.

  3. Step 1: the six faces of a dice, numbered 1 to 6. Step 2: the faces 2, 4 and 6 are hatched. Step 3: a label says chance 3/6 = 1/2.123456chance 3/6 = 1/2

    3 Write the chance as a fraction: 3 out of 6 is 3/6, which is 1/2 in simplest form.

The rule is: probability = outcomes that count ÷ all outcomes. It works when every outcome is equally likely.

Worked example

A bag holds 3 red beads and 5 green beads of the same size. You pick one without looking. What is the chance that it is red?

  1. There are 3 + 5 = 8 beads in all. Each bead is equally likely to be picked.
  2. Three of the beads are red. These are the outcomes that count.
  3. The chance of red is 3 out of 8, which is 3/8.

Answer: 3/8

A bar of 8 equal parts, one for each bead. 3 parts are shaded for the red beads. The caption says 3 red beads in 8.3 red beads in 8
Each part is one bead. The shaded parts count. The chance of red is the shaded parts out of all the parts.

Now ask the opposite: what is the chance that something does not happen? Every outcome either counts or does not count. So the two chances add up to 1. The chance of "not" is 1 take away the chance of the event.

Worked example

Use the same bag of 3 red and 5 green beads. What is the chance that the bead is not red?

  1. The chance of red is 3/8.
  2. The beads that are not red are 8 − 3 = 5.
  3. The chance of not red is 5/8. The two chances add up to 1: 3/8 + 5/8 = 1.

Answer: 5/8

A bar of 8 equal parts, one for each bead. 5 parts are shaded for the beads that are not red. The caption says 5 beads that are not red.5 beads that are not red
The shaded parts are now the 5 beads that are not red: 5 parts out of 8.

Some events have more than one stage, such as tossing two coins. List every outcome in a grid. Write one coin down the side and the other across the top. Each cell is one outcome.

A grid of the outcomes of two coins. The labels are first and second coin. The cells are HH, HT, TH, TT.second coinfirstHTHTHHHTTHTT
Two coins give 4 outcomes. HT and TH are different: the first coin is heads in HT and tails in TH.

Worked example

Two coins are tossed. What is the chance of at least one head?

  1. List the outcomes: HH, HT, TH and TT.
  2. There are 2 × 2 = 4 outcomes in all, and each is equally likely.
  3. The outcomes with at least one head are HH, HT and TH. That is 3 outcomes.
  4. The chance is 3/4, which is 3 ÷ 4 = 0.75.

Answer: 3/4

A grid of the outcomes of two coins. The labels are first and second coin. The cells HH, HT and TH are hatched because they have at least one head. The cell TT is not hatched.second coinfirstHTHTHHHTTHTT
Three cells are hatched out of four. The one plain cell, TT, is the chance of no head: 1/4.

See it three ways

With things

Roll a dice. List what can happen. Count the faces that show 5 or 6.

As a picture

The six faces of a dice, numbered 1 to 6. The faces 5 and 6 are hatched.123456

In symbols

2 faces out of 6 is 2/6. In simplest form, 2/6 = 1/3.

Try it

List every outcome

You need: paper and a pencil, a coin, if you have one.

  1. Write H for heads and T for tails.
  2. Write every pair for two coins by putting H and then T after each. You get HH, HT, TH and TT.
  3. Count the outcomes. There are 4.
  4. Count the outcomes with exactly one head. There are 2.

What you will see: Two of the four outcomes have exactly one head, HT and TH. The chance of exactly one head is 2 out of 4, which is 1/2.

If you cannot try it, this is what happens.
A grid of the outcomes of two coins. The labels are first and second coin. The cells are HH, HT, TH and TT. The cells HT and TH are hatched because they have exactly one head.second coinfirstHTHTHHHTTHTT

A wrong idea many people have

Two coins have three outcomes: two heads, two tails, or one of each. So each has a chance of 1/3.

Why it feels right: Three outcomes sound like a fair list, and "one of each" is one thing to say.

The test that breaks it: Toss two coins 100 times and tally. "One of each" usually comes up about 50 times, as often as the other two put together.

What is true: "One of each" is two outcomes, HT and TH. The four outcomes HH, HT, TH and TT are equally likely, so each has a chance of 1/4.

Key words

outcome
one possible result, such as a 4 on a dice.
equally likely
every outcome has the same chance.

Back to your guess

The answer is C: The chance of an even number is 1/2.

A dice has 6 faces, and 3 of them are even: 2, 4 and 6. The chance is 3 out of 6, which is 3/6. In simplest form that is 1/2.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. You roll a dice. What is the chance of a 5?

    Tap your answer.

    1. Yes, that is right. There is 1 face with a 5 and there are 6 faces in all. The chance is 1/6.

    2. Not this one. A dice has 6 faces, not 5. There are 6 equally likely outcomes in all. Go to M3.7.7. Try another.

    3. Not this one. That is the chance of not rolling a 5. One face out of 6 shows a 5. Go to M3.7.7. Try another.

    See the answer

    1/6

    There is 1 face with a 5 and there are 6 faces in all. The chance is 1/6.

    • If you chose B: A dice has 6 faces, not 5. There are 6 equally likely outcomes in all. Go to M3.7.7.
    • If you chose C: That is the chance of not rolling a 5. One face out of 6 shows a 5. Go to M3.7.7.

  2. A bag holds 2 red, 6 blue and 2 yellow beads of the same size. What is the chance of picking blue?

    Tap your answer.

    1. Not this one. You compared blue with the other beads. A chance is the beads that count out of all the beads. It cannot be more than 1. Go to M3.7.7. Try another.

    2. Yes, that is right. There are 2 + 6 + 2 = 10 beads. Six are blue, so the chance is 6/10. In simplest form that is 3/5.

    3. Not this one. There are 3 colours, but the colours are not equally likely. Count the beads: 6 blue out of 10. Go to M3.7.6. Try another.

    See the answer

    3/5

    There are 2 + 6 + 2 = 10 beads. Six are blue, so the chance is 6/10. In simplest form that is 3/5.

    • If you chose A: You compared blue with the other beads. A chance is the beads that count out of all the beads. It cannot be more than 1. Go to M3.7.7.
    • If you chose C: There are 3 colours, but the colours are not equally likely. Count the beads: 6 blue out of 10. Go to M3.7.6.

  3. You roll a dice. What is the chance of not rolling a 6?

    Tap your answer.

    1. Not this one. That is the chance of rolling a 6. The question asks for the chance of not rolling a 6. Go to M3.7.7. Try another.

    2. Not this one. A chance is never more than 1. Five faces count, out of 6 faces in all. Go to M3.7.7. Try another.

    3. Yes, that is right. The chance of a 6 is 1/6. The chance of not a 6 is 1 take away 1/6, which is 5/6.

    See the answer

    5/6

    The chance of a 6 is 1/6. The chance of not a 6 is 1 take away 1/6, which is 5/6.

    • If you chose A: That is the chance of rolling a 6. The question asks for the chance of not rolling a 6. Go to M3.7.7.
    • If you chose B: A chance is never more than 1. Five faces count, out of 6 faces in all. Go to M3.7.7.

  4. Two coins are tossed. Look at the grid. What is the chance of two tails?

    A grid of the outcomes of two coins. The labels are first and second coin. The cells are HH, HT, TH, TT. The cell TT is hatched.second coinfirstHTHTHHHTTHTT
    Tap your answer.

    1. Yes, that is right. One outcome out of the 4 is TT, so the chance is 1/4.

    2. Not this one. HT and TH are two different outcomes, so there are 4 equally likely outcomes, not 3. Go to M3.7.7. Try another.

    3. Not this one. Half would be 2 cells out of 4. Only 1 cell, TT, shows two tails. Go to M3.7.7. Try another.

    See the answer

    1/4

    One outcome out of the 4 is TT, so the chance is 1/4.

    • If you chose B: HT and TH are two different outcomes, so there are 4 equally likely outcomes, not 3. Go to M3.7.7.
    • If you chose C: Half would be 2 cells out of 4. Only 1 cell, TT, shows two tails. Go to M3.7.7.

  5. A spinner has 5 equal sections numbered 1 to 5. What is the chance of a number greater than 3?

    Tap your answer.

    1. Not this one. That counts 1, 2 and 3, the numbers that are 3 or less. "Greater than 3" means 4 and 5. Go to M3.7.7. Try another.

    2. Yes, that is right. The numbers greater than 3 are 4 and 5, so 2 sections count. There are 5 sections in all. The chance is 2/5.

    3. Not this one. You compared the 2 numbers that win with the 3 that do not. Divide by all 5 sections. Go to M3.7.7. Try another.

    See the answer

    2/5

    The numbers greater than 3 are 4 and 5, so 2 sections count. There are 5 sections in all. The chance is 2/5.

    • If you chose A: That counts 1, 2 and 3, the numbers that are 3 or less. "Greater than 3" means 4 and 5. Go to M3.7.7.
    • If you chose C: You compared the 2 numbers that win with the 3 that do not. Divide by all 5 sections. Go to M3.7.7.

  6. A bag holds 4 red beads and 8 green beads of the same size. What is the chance of picking red? Write it in simplest form.

    See the answer

    1/3

    There are 4 + 8 = 12 beads. Four are red, so the chance is 4/12. In simplest form that is 1/3.

  7. A bag holds 5 beads of the same size and 1 is black. What is the chance of not picking black? Write it as a decimal.

    See the answer

    0.8

    The chance of black is 1/5, which is 0.2. The chance of not black is 1 − 0.2 = 0.8.

  8. From an earlier chapter

    Review: write the probability 3/5 as a decimal.

    See the answer

    0.6

    Divide the top by the bottom: 3 ÷ 5 = 0.6.

Keep it

Copy this into your exercise book.

Probability = outcomes that count ÷ all outcomes, when every outcome is equally likely. The chance of "not" is 1 take away the chance. Two coins have 4 outcomes: HH, HT, TH, TT.

Draw: Draw the 4 outcomes of two coins in a grid. Hatch the cells with exactly one head.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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