Grade by Grade
D. Data · Data and Probability

Probability as a Fraction on the Probability Line

D2.1About 45 minutes
Learning goals
  • I can write the probability of an event as a fraction.
  • I can place probabilities from 0 to 1 on a probability line.
  • I can compare probabilities to decide which event is more likely.
  • I can use probability to make predictions and informed decisions.

Words to know

probability
A number from 0 to 1 that tells how likely an event is.
outcome
One possible result, like rolling a 4 on a die.
event
One outcome or a group of outcomes you are interested in, like rolling an even number.
favourable outcome
An outcome that makes the event happen.
equally likely outcomes
Outcomes that all have the same chance, like the 6 faces of a fair die.

Remember from Grade 4

You placed events on a probability line from impossible to certain, using words like unlikely, equally likely and likely. Now you'll give each event an exact number: a fraction from 0 to 1.

Which spinner would you choose?

At the school fun fair, you win a prize if the spinner lands on green. Look at Spinner A. Is winning likely? How likely? A fraction can tell us exactly.

Spinner AGreenBlueYellow8 equal sections:4 green, 3 blue, 1 yellow
Spinner A has 8 equal sections: 4 green, 3 blue and 1 yellow.

Probability as a fraction

When all the outcomes are equally likely:

Probability of an event = number of favourable outcomes ÷ number of possible outcomes

Spinner A has 8 equal sections, so there are 8 equally likely outcomes.

  • P(green) = 4/8 = 1/2
  • P(blue) = 3/8
  • P(yellow) = 1/8
  • P(red) = 0/8 = 0 (there is no red: impossible)
  • P(green, blue or yellow) = 8/8 = 1 (certain)

"P(green)" is a short way to write "the probability of landing on green."

The probability line with numbers

  • 0 means impossible.
  • 1 means certain.
  • 1/2 means equally likely (an even chance).
  • Fractions less than 1/2 are unlikely, and fractions greater than 1/2 are likely.
01/82/83/84/85/86/87/81impossibleequally likelycertainABCDE
Spinner A events: A = red (0), B = yellow (1/8), C = blue (3/8), D = green (4/8 = 1/2), E = green, blue or yellow (1).
Worked example

Example 1: Rolling a die

A regular die has the numbers 1 to 6. Find each probability and describe it: (a) rolling a 4, (b) rolling an even number, (c) rolling a number greater than 4, (d) rolling a 7.

  1. There are 6 equally likely outcomes: 1, 2, 3, 4, 5, 6.
  2. (a) Only one outcome is 4. P(4) = 1/6. Unlikely.
  3. (b) Even numbers: 2, 4, 6. P(even) = 3/6 = 1/2. Equally likely.
  4. (c) Greater than 4: 5, 6. P(greater than 4) = 2/6 = 1/3. Unlikely, but more likely than rolling a 4.
  5. (d) There is no 7. P(7) = 0. Impossible.

1/6, 1/2, 1/3, 0

Worked example

Example 2: Make a prediction

You spin Spinner A 40 times. About how many times do you predict it will land on each colour?

  1. P(green) = 1/2. Half of 40 is 20.
  2. P(blue) = 3/8. 40 ÷ 8 = 5, so 1/8 of 40 is 5, and 3/8 of 40 is 3 × 5 = 15.
  3. P(yellow) = 1/8. 1/8 of 40 is 5.
  4. Check: 20 + 15 + 5 = 40. ✓

About 20 green, 15 blue and 5 yellow. Real results will probably be a little different.

Worked example

Example 3: Make an informed decision

You win a prize on green. Spinner A has P(green) = 1/2. A bag has 7 green marbles and 3 red marbles, and you win if you pull out green without looking. Which game should you play?

  1. Bag: 10 marbles, 7 green. P(green) = 7/10.
  2. Spinner A: P(green) = 1/2 = 5/10.
  3. 7/10 is greater than 5/10.

Play the marble game. Its probability of winning (7/10) is greater.

A fraction is not a promise

P(green) = 1/2 doesn't mean every second spin will be green. It means that over many spins, about half of them should be green. In the next lesson, you'll test this with experiments.

Just for fun

What if Spinner A had 8 green sections? Then P(green) = 8/8 = 1, everyone would win, and the prize booth would run out of prizes before lunch!

Hands-on activity

Build a spinner

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

  1. Trace a circle and cut it out. (A grown-up can help with the scissors.)
  2. Fold it in half, then in half again, then in half again. Unfold: you have 8 equal sections.
  3. Colour 4 sections green, 3 blue and 1 yellow, like Spinner A.
  4. Put the paper clip at the centre and hold it in place with the tip of the pencil. Flick the clip to spin.
  5. Write P(green), P(blue) and P(yellow) as fractions.
  6. Design challenge: colour a new 8-section spinner so that P(yellow) = 1/4 and P(blue) = 1/2. (Keep your spinners for the next lesson!)
Spinner BGreenYellowBlue6 equal sections:3 green, 2 yellow, 1 blue
Use Spinner B for the Your turn questions. It has 6 equal sections: 3 green, 2 yellow and 1 blue.

Your turn

1

Spinner B: what is P(green)?

2

Spinner B: what is P(yellow)?

3

Spinner B: what is P(blue)?

4

Spinner B: what is P(red)? Describe it in words.

5

Spinner B: which colour is least likely? Where does it go on the probability line?

6

A bag has 2 red, 5 blue and 3 green marbles. What is P(blue)?

7

You spin Spinner B 30 times. About how many times do you predict green?

More practice: warm-up

1

What probability means impossible?

2

What probability means certain?

3

A coin is flipped. What is P(heads)?

4

A die is rolled. What is P(3)?

5

Is an event with probability 4/5 likely or unlikely?

More practice: on your own

1

A die is rolled. What is P(a number less than 3)?

2

A bag has 4 red and 6 yellow marbles. Which is more likely, red or yellow? Give both probabilities.

3

Order these from least likely to most likely: P = 3/4, P = 1/4, P = 1/2.

4

Spinner A is spun 80 times. Predict how many times it lands on blue.

5

The letters of the word CANADA are put on 6 cards. You pick one without looking. What is P(A)?

Challenge

1

Design an 8-section spinner where P(yellow) = 1/4 and P(blue) = 1/2. The rest are green. How many sections of each colour? What is P(green)?

2

Game 1: you win if a die shows 5 or 6. Game 2: you win if a spinner with 4 equal sections, 1 of them gold, lands on gold. Which game gives you the better chance? Explain.

3

A bag has 12 marbles. P(red) = 1/3 and P(blue) = 1/2. The rest are green. How many marbles of each colour are there?

Check your understanding

Quiz

1.

A spinner has 8 equal sections: 4 green, 3 blue, 1 yellow. What is P(blue)?

2.

Which probability means impossible?

3.

A die is rolled. What is P(even number)?

4.

Which event is most likely?

5.

A spinner has P(green) = 1/4. It is spun 20 times. About how many greens should you predict?

6.

Where does P = 1/2 go on a probability line?

For parents and teachers

What this lesson teaches: using fractions to express probabilities, placing them on a probability line from 0 (impossible) to 1 (certain), and using them to make predictions and informed decisions (D2.1). New in Grade 5: the probability line now uses fractions instead of only words.

How to help at home:

  • Use dice, coins, spinners, or a bag of coloured objects to ask "What's the probability?"
  • Ask decision questions: "Which game gives you the better chance of winning?"
  • Practise finding a fraction of a number for predictions (3/8 of 40: find 1/8, then multiply by 3).

Common difficulties:

  • Writing a probability as "3 to 5" (comparing favourable to unfavourable) instead of 3/8 (favourable out of all).
  • Counting sections without checking that they are equal. The fraction rule only works for equally likely outcomes.
  • Thinking P = 1/2 means it will happen exactly every second time.
  • Comparing fractions with different denominators. Use equivalent fractions with the same denominator.