Theoretical probability | Year 7

01 / LEARN

The idea, step by step

Build the probability fraction

When elementary outcomes are equally likely, divide the favourable count by the total possible count. Favourable means “meets the event description”, not “a good result”. Write the outcomes down to make both counts visible.

For an even result on a fair die, the favourable faces are 2, 4 and 6. There are three favourable results among six, so the chance is $3 / 6 = 1 / 2$ . Do not divide by the unfavourable count.

P (E) = \frac{favourable outcomes}{all equally likely outcomes}

Represent and check the answer

The same chance can be written as $1 / 4$ , $0.25$ or $25 %$ . Fractions keep simple chances exact. To compare, use common denominators or familiar decimal equivalents.

A probability cannot be negative or greater than 1. An impossible event has chance 0; a certain event has chance 1. If you get a fraction greater than 1, check for double counting or a wrong denominator.

A common trap

Using favourable outcomes divided by unfavourable outcomes.

Instead: The denominator includes every possible equally likely outcome, including the favourable ones.

02 / WORKED EXAMPLES

See the thinking

Example 1

A number-card condition

One card is selected at random from cards numbered 1 to 12. Find the probability of a multiple of 3.

The favourable numbers are 3, 6, 9 and 12.
Four of the twelve cards are favourable.
Simplify $4 / 12$ by dividing both parts by 4.

1 / 3

Example 2

An event described by “not”

A bag has 4 red, 3 green and 3 yellow equal-size counters. Find the chance that a random selection is not red.

There are $4 + 3 + 3 = 10$ counters.
Not red means green or yellow: $3 + 3 = 6$ .
Calculate $6 / 10 = 3 / 5 = 0.6$ .

3 / 5 = 60 %

Check your understanding

One of six cards numbered 1 to 6 is selected at random. What is the chance of a number greater than 4?

Show answer

$2 / 6 = 1 / 3$ .

Only 5 and 6 satisfy the condition, out of all six cards.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A fair die is rolled. Find the probability of a 5.
2 marks
A fair die is rolled. Find the probability of an odd number.
2 marks
A card is selected at random from cards numbered 1 to 10. Find the probability of a factor of 10.
2 marks
A counter is chosen at random from a bag with 7 white and 3 black equal-size counters. Give the chance of black as a fraction, decimal and percentage.
2 marks
A fair die is rolled. Find the chance of a number less than 1 and the chance of a number at most 6.
2 marks
Which is greater: the chance of blue when a counter is chosen at random from 3 blue and 5 red equal-size counters, or an even result on a fair die?
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A fair die is rolled. Find the probability of a 5.
2 marks
One face is labelled 5.
There are six equally likely faces.
$1 / 6$ .
A fair die is rolled. Find the probability of an odd number.
2 marks
The odd results are 1, 3 and 5.
Three of the six results are favourable.
$3 / 6 = 1 / 2$ .
A card is selected at random from cards numbered 1 to 10. Find the probability of a factor of 10.
2 marks
The factors in the set are 1, 2, 5 and 10.
Four of ten cards are favourable.
$4 / 10 = 2 / 5$ .
A counter is chosen at random from a bag with 7 white and 3 black equal-size counters. Give the chance of black as a fraction, decimal and percentage.
2 marks
There are ten counters altogether.
Three favourable counters give $3 / 10$ .
$3 / 10 = 0.3 = 30 %$ .
A fair die is rolled. Find the chance of a number less than 1 and the chance of a number at most 6.
2 marks
No possible face is less than 1.
All six possible faces are at most 6.
Less than 1: 0. At most 6: 1.
Which is greater: the chance of blue when a counter is chosen at random from 3 blue and 5 red equal-size counters, or an even result on a fair die?
3 marks
Blue chance is $3 / 8$ .
The even chance is $1 / 2 = 4 / 8$ .
$4 / 8 > 3 / 8$ .
An even result on the die.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Year 7 single-stage sample spaces, equally likely outcomes, probabilities, expected frequencies, chance experiments, simulations and relative frequencies.

Official curriculum source ↗

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