Grade by Grade
D. Data · Data and Probability

Likelihood and the Probability Line

D2.1About 45 minutes
Learning goals
  • I can describe events using impossible, unlikely, equally likely, likely and certain.
  • I can place events on a probability line and compare how likely they are.
  • I can use likelihood to make predictions and informed decisions.

Words to know

event
Something that might happen, like landing on green or rolling a 6.
likelihood
How likely it is that an event will happen.
probability line
A line from impossible to certain that shows how likely events are.
equally likely
Two events have the same chance of happening, like heads or tails on a coin.
prediction
A thoughtful guess about what will happen, based on what you know.

Remember from Grade 3

You described events with five words: impossible, unlikely, equally likely, likely and certain. In Grade 4 you will put events on a probability line so you can see and compare how likely they are.

Snow is likely tomorrow

The weather forecast says: "Snow is likely tomorrow." What will you wear to school? Probably your boots and snow pants! You used likelihood to make a decision. Likely doesn't mean certain. It might not snow. But it's smart to be ready.

A quiet snowy street with houses, and a pair of winter boots ready on a front step.
When snow is likely, it makes sense to have your boots ready, even though snow is not certain. AI-generated image

The probability line

A probability line is like a number line for likelihood.

  • Impossible is at the left end. It can never happen.
  • Certain is at the right end. It will definitely happen.
  • Equally likely is right in the middle: an even chance.
  • Unlikely is between impossible and equally likely.
  • Likely is between equally likely and certain.

The farther right an event is, the more likely it is.

ImpossibleEqually likelyCertainUnlikelyLikely
A probability line: impossible, unlikely, equally likely, likely, certain, from left to right.
Spinner ASpinner B
Spinner A: three quarters green, one quarter yellow. Spinner B: half green, half blue.
Worked example

Example 1: Spinner A on the probability line

Spin Spinner A. Describe each event and place it on a probability line.

A: land on green. B: land on yellow. C: land on blue. D: land on green or yellow.

  1. Green covers 3 of the 4 equal parts, so it is likely.
  2. Yellow covers only 1 of the 4 parts, so it is unlikely.
  3. There is no blue on Spinner A, so landing on blue is impossible.
  4. The whole spinner is green or yellow, so D is certain.

See the probability line below. A is likely, B is unlikely, C is impossible and D is certain.

ImpossibleEqually likelyCertainUnlikelyLikelyCBAD
Spinner A events: C at impossible, B at unlikely, A at likely, D at certain.
Worked example

Example 2: Equally likely

On Spinner B, are you more likely to land on green or on blue?

  1. Green covers half of Spinner B. Blue covers the other half.
  2. The two parts are the same size, so each colour has the same chance.

Green and blue are equally likely. Both go in the middle of the probability line.

🔴🔴🔴🔴🔴🔴🔵🔵
A bag with 6 red marbles and 2 blue marbles.
Worked example

Example 3: Picking from a bag

You reach into the bag without looking and take one marble. Describe each event: picking red, picking blue, picking yellow.

  1. There are 6 red and 2 blue, so 8 marbles in all.
  2. Red: 6 of the 8 marbles. Likely.
  3. Blue: only 2 of the 8. Unlikely.
  4. Yellow: there are no yellow marbles. Impossible.

Red is likely, blue is unlikely, yellow is impossible. Red is more likely than blue.

In between the words

Events don't have to sit right on a word. Think of a regular die with the numbers 1 to 6. Rolling a 6 is unlikely: only 1 of the 6 numbers works. Rolling a 1 or a 2 is also unlikely, but 2 numbers work, so it is a bit more likely than rolling a 6. It goes a little farther right on the line.

Worked example

Example 4: Place the dice events

Roll a die numbered 1 to 6. Place these events on a probability line.

A: roll a 6. B: roll an even number. C: roll a number less than 7. D: roll a 0. E: roll a number greater than 1.

  1. A: 1 of 6 numbers works. Unlikely, quite close to impossible.
  2. B: 2, 4, 6 are even. 1, 3, 5 are odd. 3 of 6 numbers work: equally likely.
  3. C: every number on the die is less than 7. Certain.
  4. D: there is no 0 on the die. Impossible.
  5. E: 2, 3, 4, 5, 6 work. 5 of 6 numbers: likely, quite close to certain.

From least to most likely: D, A, B, E, C.

ImpossibleEqually likelyCertainUnlikelyLikelyDABEC
Dice events: D at impossible, A near impossible, B at equally likely, E near certain, C at certain.
Worked example

Example 5: Predict and decide

In a game, you win a point if the spinner lands on green. You can choose Spinner A or Spinner B. Which should you choose? If you spin Spinner A 20 times, about how many times would you predict green?

  1. Spinner A: green is 3 of 4 parts, so likely. Spinner B: green is half, so equally likely.
  2. Green is more likely on Spinner A, so choose Spinner A. That's an informed decision.
  3. In 20 spins, green should come up most of the time. 3 out of every 4 spins would be 15 out of 20.
  4. So predict about 15 greens. The real result might be 13, or 16, or something else. Likely is not certain!

Choose Spinner A. Predict about 15 greens out of 20 spins.

Likely is not certain

Even a likely event sometimes doesn't happen, and an unlikely event sometimes does. If you spin only a few times, the results can surprise you. The more times you spin, the closer your results usually get to what you predicted.

Just for fun

What's the probability that a cow will jump over the moon? Impossible.

What's the probability that you just read this joke? Certain!

Hands-on activity

Paper clip spinner

You need: paper, a small plate or lid to trace, a ruler, pencil crayons, a pencil and a paper clip.

  1. Trace a circle. Use the ruler to divide it into 4 equal parts (draw a + through the centre).
  2. Colour 3 parts one colour and 1 part another colour.
  3. Put the pencil point on the centre, through the end of the paper clip. Flick the paper clip to spin. (Keep the pencil point pointing down, away from your face.)
  4. Predict: in 20 spins, how many times will it land on each colour? Mark both colours on a probability line.
  5. Spin 20 times and keep a tally.
  6. Compare: were your results close to your prediction?

Math talk: "If I wanted my two colours to be equally likely, how would I change the spinner?"

Spinner C
Spinner C: half blue, one quarter green, one quarter yellow. Use it for questions 1 to 4.

Your turn

1

Spinner C: how likely is landing on blue?

2

Spinner C: how likely is landing on green?

3

Spinner C: how likely is landing on red?

4

Spinner C: how likely is landing on blue, green or yellow?

5

The bag has 6 red and 2 blue marbles. How likely is picking blue?

6

Where does certain go on a probability line?

7

Roll a die numbered 1 to 6. How likely is rolling a number less than 5?

More practice: warm-up

1

Roll a die numbered 1 to 6. How likely is rolling a 9?

2

Flip a coin. How likely is heads?

3

A spinner is all green. How likely is landing on green?

4

A bag has 9 red cubes and 1 blue cube. How likely is picking red?

5

Which end of the probability line is impossible?

More practice: on your own

1

Roll a die numbered 1 to 6. Order these events from least likely to most likely: A: roll a 6. B: roll an even number. C: roll a 7.

2

Design a bag of 10 cubes where picking red is likely but not certain.

3

A bag has 5 red and 5 blue cubes. How likely is picking red?

4

Lily spun Spinner A (3 parts green, 1 part yellow) 4 times and got yellow 3 times. She says the spinner is broken. Do you agree?

5

The forecast says rain is likely this afternoon. What is an informed decision?

Challenge

1

Design a spinner with 4 equal parts where green is unlikely, blue is likely and yellow is impossible.

2

In a game you can choose: win if you roll an even number, or win if you roll a number less than 3. Which should you choose? Explain.

3

A bag has 3 red marbles and 1 blue marble. You take out a red marble and keep it out. Is picking red now more likely or less likely than before? Where does it move on the probability line?

Check your understanding

Quiz

1.

Where does equally likely go on a probability line?

2.

A spinner has 4 equal parts: 3 blue and 1 yellow. Landing on yellow is...

3.

Roll a die numbered 1 to 6. Rolling a number less than 7 is...

4.

A bag has 2 red and 8 blue cubes. Which is more likely?

5.

Which event goes farthest right on a probability line?

6.

You win on green. Spinner A is 3 parts green and 1 part yellow. Spinner B is half green and half blue. Which spinner is the better choice?

For parents and teachers

What this lesson teaches: using the words impossible, unlikely, equally likely, likely and certain; placing events on a probability line (new in Grade 4); comparing likelihoods; and using them to make predictions and informed decisions (D2.1).

How to help at home:

  • Use the language in daily life: "Is it likely we'll see a squirrel on our walk?" "Is it certain the bus comes at 8:05?"
  • Play board games with dice and ask which rolls are more likely.
  • Talk about weather forecasts and the decisions you make because of them.

Common difficulties:

  • Thinking "likely" means "will happen". Likely events sometimes don't happen.
  • Ignoring the size of spinner parts, or the number of each colour in a bag.
  • Expecting experiment results to match the prediction exactly, especially with only a few tries.