Chapter 6 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

M3.7.6 Chapter 6

The language of chance

A bag holds 1 black bead and 9 white beads. How likely is it that you pick black?

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

  • Place events on a scale from impossible to certain
  • Write a probability as a fraction or decimal from 0 to 1

An event is something that may happen, such as "a coin lands on heads". Some events cannot happen. Some must happen. Most are in between.

The probability of an event is a number from 0 to 1 that says how likely it is. We put these numbers on a line, with the five words in their places.

Draw it first

Build the probability line

  1. Step 1: a line from 0 to 1 with ticks at 0, 1/4, 1/2, 3/4 and 1. Step 2: the words impossible, even chance, certain, unlikely and likely are placed on the line. Step 3: four events are placed on the line. a 7: 0, a 6: 1/6, heads: 1/2 and under 7: 1.01/41/23/41

    1 Draw a line. Write 0 at the left end and 1 at the right end. Mark the quarters on it.

  2. Step 1: a line from 0 to 1 with ticks at 0, 1/4, 1/2, 3/4 and 1. Step 2: the words impossible, even chance, certain, unlikely and likely are placed on the line. Step 3: four events are placed on the line. a 7: 0, a 6: 1/6, heads: 1/2 and under 7: 1.01/41/23/41impossiblecertainevenchanceunlikelylikely

    2 Write the words. Impossible is at 0. Even chance is half way. Certain is at 1. Unlikely is between 0 and the middle. Likely is between the middle and 1.

  3. Step 1: a line from 0 to 1 with ticks at 0, 1/4, 1/2, 3/4 and 1. Step 2: the words impossible, even chance, certain, unlikely and likely are placed on the line. Step 3: four events are placed on the line. a 7: 0, a 6: 1/6, heads: 1/2 and under 7: 1.01/41/23/41a 7: 0a 6: 1/6heads: 1/2under 7: 1impossiblecertainevenchanceunlikelylikely

    3 Put each event where its probability says. Rolling a 7 on a dice is at 0, and a coin showing heads is half way.

Here are the meanings. Impossible is 0: it can never happen. Certain is 1: it always happens. Even chance is 1/2, or 0.5: it is as likely to happen as not. A dice is the object with faces 1 to 6.

The six faces of a dice, with 1 to 6 pips and the numbers 1, 2, 3, 4, 5 and 6 written under them. The caption says one dice.123456one dice
A dice has six faces. Each face is equally likely to land on top.

A probability is never below 0 and never above 1. You may write it as a fraction, such as 1/4, or as a decimal, such as 0.25. They are the same number.

Worked example

A bag holds 4 beads of the same size: 1 black and 3 white. You pick one without looking. Say how likely black is, in words, as a fraction and as a decimal.

  1. The beads are the same size, so each bead is equally likely to be picked.
  2. One bead out of 4 is black. The chance of black is 1/4.
  3. This is less than 0.5, so black is unlikely. As a decimal, 1 ÷ 4 = 0.25.

Answer: Unlikely, 1/4, 0.25

A bar of 4 equal parts, one shaded for the black bead. The caption says 1 black bead in 4.1 black bead in 4
The shaded part is the black bead. Black is 1 part out of 4, which is 1/4.

Worked example

Use the same bag. How likely is it that you pick a white bead? Say it in words, then as a fraction and a decimal.

  1. Three of the 4 beads are white. The chance of white is 3/4.
  2. Divide the top by the bottom: 3 ÷ 4 = 0.75.
  3. It is more than 0.5, so white is likely. It sits at 0.75 on the line.

Answer: Likely, 3/4, 0.75

A line from 0 to 1 with ticks at 0, 0.25, 0.5, 0.75 and 1. The events are black: 0.25 and white: 0.75. Under it the words impossible, even chance, certain, unlikely and likely.00.250.50.751black: 0.25white: 0.75impossiblecertainevenchanceunlikelylikely
Black is at 0.25, in the unlikely part. White is at 0.75, in the likely part.

See it three ways

With things

A bag holds 1 black bead and 1 white bead. Shake it and pick one without looking. Each bead is equally likely.

As a picture

A line from 0 to 1 with ticks at 0, 0.25, 0.5, 0.75 and 1. One event is black: 0.5, at the middle of the line. Under the line are the words impossible, even chance, certain, unlikely and likely.00.250.50.751black: 0.5impossiblecertainevenchanceunlikelylikely

In symbols

1 bead out of 2 is black. 1 ÷ 2 = 0.5. This is an even chance.

Try it

Put events on a line

You need: paper and a pencil, a ruler.

  1. Draw a line 20 cm long. Write 0 at the left end and 1 at the right end.
  2. Write four events on small pieces of paper. A dice shows 7. A dice shows 6. A coin shows heads. A dice shows a number under 6.
  3. Decide where each event goes. Put the paper on the line.
  4. Check each one with the words: impossible, unlikely, even chance, likely, certain.

What you will see: A dice showing 7 goes at 0. A 6 is at 1/6, near 0. Heads is at 1/2. A number under 6 is 5 faces out of 6. It is at 5/6, near 1.

If you cannot try it, this is what happens.
A line from 0 to 1 with ticks at 0, 1/4, 1/2, 3/4 and 1. Under it the words impossible, even chance, certain, unlikely and likely. The events are a 7: 0, a 6: 1/6, heads: 1/2, under 6: 5/6.01/41/23/41a 7: 0a 6: 1/6heads: 1/2under 6: 5/6impossiblecertainevenchanceunlikelylikely

A wrong idea many people have

There are two things that can happen, black or white, so each has a chance of 1/2.

Why it feels right: It seems fair to share the chance equally between two answers.

The test that breaks it: A bag holds 1 black bead and 9 white beads. Pick one 100 times, putting it back each time. Black usually comes up about 10 times, far fewer than half.

What is true: Chance is shared equally only when the outcomes are equally likely. Count the beads to find the chance.

Key words

event
something that may happen.
probability
a number from 0 to 1 that says how likely an event is.
impossible
it can never happen. Its probability is 0.
certain
it always happens. Its probability is 1.
even chance
as likely to happen as not. Its probability is 0.5.

Back to your guess

The answer is C: Unlikely

Only 1 bead out of 10 is black. Black can happen, but it will not happen most of the time. That is unlikely. In this chapter you will give each of these words a number from 0 to 1.

Check yourself

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

  1. You roll a dice. How likely is it that the dice shows 7?

    Tap your answer.

    1. Yes, that is right. A dice shows 1, 2, 3, 4, 5 or 6. A 7 can never show, so the probability is 0.

    2. Not this one. Unlikely means it can still happen. A dice has the faces 1 to 6 and no face with 7, so it cannot happen. Go to M3.7.6. Try another.

    3. Not this one. An even chance is a probability of 0.5. A 7 can never show, so its probability is 0. Go to M3.7.6. Try another.

    See the answer

    Impossible

    A dice shows 1, 2, 3, 4, 5 or 6. A 7 can never show, so the probability is 0.

    • If you chose B: Unlikely means it can still happen. A dice has the faces 1 to 6 and no face with 7, so it cannot happen. Go to M3.7.6.
    • If you chose C: An even chance is a probability of 0.5. A 7 can never show, so its probability is 0. Go to M3.7.6.

  2. You roll a dice. How likely is it that the dice shows a number less than 7?

    Tap your answer.

    1. Not this one. Likely means it may fail to happen. Every face of a dice is less than 7, so it always happens. Go to M3.7.6. Try another.

    2. Yes, that is right. All six faces, 1 to 6, are less than 7. It always happens, so the probability is 1.

    3. Not this one. Only half the faces would give an even chance. Here all 6 faces count. Go to M3.7.6. Try another.

    See the answer

    Certain

    All six faces, 1 to 6, are less than 7. It always happens, so the probability is 1.

    • If you chose A: Likely means it may fail to happen. Every face of a dice is less than 7, so it always happens. Go to M3.7.6.
    • If you chose C: Only half the faces would give an even chance. Here all 6 faces count. Go to M3.7.6.

  3. Which of these numbers cannot be a probability?

    Tap your answer.

    1. Not this one. The number 0.3 is between 0 and 1, so it can be a probability. Go to M3.7.6. Try another.

    2. Not this one. The number 1 is the biggest probability. It means certain. Go to M3.7.6. Try another.

    3. Yes, that is right. A probability is never above 1, because 1 means certain. Nothing is more than certain.

    See the answer

    1.2

    A probability is never above 1, because 1 means certain. Nothing is more than certain.

    • If you chose A: The number 0.3 is between 0 and 1, so it can be a probability. Go to M3.7.6.
    • If you chose B: The number 1 is the biggest probability. It means certain. Go to M3.7.6.

  4. Which event on the line has an even chance?

    A line from 0 to 1 with ticks at 0, 1/4, 1/2, 3/4 and 1. Under it the words impossible, even chance, certain, unlikely and likely. The events are a 7: 0, a 6: 1/6, heads: 1/2, under 6: 5/6.01/41/23/41a 7: 0a 6: 1/6heads: 1/2under 6: 5/6impossiblecertainevenchanceunlikelylikely
    Tap your answer.

    1. Yes, that is right. Heads is half way along the line, at 1/2. That is an even chance.

    2. Not this one. A 6 is at 1/6, close to 0. That is unlikely, not an even chance. Go to M3.7.6. Try another.

    3. Not this one. Under 6 is at 5/6, close to 1. That is likely, not an even chance. Go to M3.7.6. Try another.

    See the answer

    heads

    Heads is half way along the line, at 1/2. That is an even chance.

    • If you chose B: A 6 is at 1/6, close to 0. That is unlikely, not an even chance. Go to M3.7.6.
    • If you chose C: Under 6 is at 5/6, close to 1. That is likely, not an even chance. Go to M3.7.6.

  5. Write the probability 1/4 as a decimal.

    Tap your answer.

    1. Not this one. You wrote the 1 and the 4 side by side. The fraction 1/4 means 1 divided by 4. That is less than 1. Go to M3.3.7. Try another.

    2. Yes, that is right. Divide the top by the bottom: 1 ÷ 4 = 0.25.

    3. Not this one. That is 4 tenths, which is 2/5. For 1/4, work out 1 ÷ 4. Go to M3.3.7. Try another.

    See the answer

    0.25

    Divide the top by the bottom: 1 ÷ 4 = 0.25.

    • If you chose A: You wrote the 1 and the 4 side by side. The fraction 1/4 means 1 divided by 4. That is less than 1. Go to M3.3.7.
    • If you chose C: That is 4 tenths, which is 2/5. For 1/4, work out 1 ÷ 4. Go to M3.3.7.

  6. A bag holds 5 beads of the same size and 2 are red. Write the probability of picking red as a decimal.

    See the answer

    0.4

    Red is 2 out of 5 beads, which is 2/5. Divide: 2 ÷ 5 = 0.4.

  7. A bag holds 8 beads of the same size and 1 is black. Write the probability of picking black as a decimal.

    See the answer

    0.125

    Black is 1 out of 8 beads. Divide: 1 ÷ 8 = 0.125.

  8. From an earlier chapter

    Review: a line graph shows Monday at 42 books and Wednesday at 47 books. How many more books were borrowed on Wednesday?

    See the answer

    5 books

    Take away: 47 − 42 = 5.

Keep it

Copy this into your exercise book.

Probability is a number from 0 to 1. Impossible is 0. Certain is 1. Even chance is 0.5. Unlikely is below 0.5. Likely is above 0.5. Write it as a fraction or a decimal.

Draw: Draw a line from 0 to 1. Mark 0, 0.5 and 1. Write impossible, even chance and certain under them.

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

No internet tomorrow? Save the whole book as one file (968 KB). It opens on any phone.