“and” versus inclusive/exclusive “or” | Year 9

01 / LEARN

The idea, step by step

And means overlap

A and B requires both conditions. A or B normally includes A, B, and any overlap. If exactly one condition is intended, the question says but not both.

On a fair die, even results are {2,4,6} and results greater than 3 are {4,5,6}. Both conditions hold at {4,6}; at least one holds at {2,4,5,6}.

Count each region once

In the diagram, A only contains 6, both contains 4, B only contains 3, and neither contains 7. A or B therefore contains 13 people. A or B but not both contains 9.

If adding total A and total B, subtract their overlap once because it was counted twice. In probability form: $P (A or B) = P (A) + P (B) - P (A and B)$ .

A common trap

Or always means exactly one condition.

Instead: Probability uses inclusive or unless but not both is stated.

02 / WORKED EXAMPLES

See the thinking

Example 1

A die union

Roll a fair die. Find P(even or greater than 3).

Even gives {2,4,6}; greater than 3 gives {4,5,6}.
Combine without repeats: {2,4,5,6}.

4 / 6 = 2 / 3

Example 2

Exactly one group

One person is chosen at random from the displayed group of 20. Find P(A or B, but not both).

Include only A and only B.
$6 + 3 = 9$ of the 20 people meet exactly one condition.

9 / 20

Check your understanding

For a fair die, A is odd and B is greater than 4. Which outcomes are in both?

Show answer

Only 5.

The intersection of {1,3,5} and {5,6} is {5}.

03 / WORKSHEET

Your turn

6 questions · 11 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

List die results that are even and greater than 2.
1 mark
Find P(odd or a 6) for a fair die.
2 marks
One person is chosen at random from the displayed group. Use the diagram to find P(A and B).
2 marks
One person is chosen at random from the displayed group. Use the diagram to find P(A or B).
2 marks
Of 30 students, 18 play football, 12 tennis and 5 both. How many play at least one?
2 marks
Of 30 students, 18 play football, 12 play tennis and 5 play both. One student is chosen at random. Find P(neither sport).
2 marks

Read the worked solutions

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

List die results that are even and greater than 2.
1 mark
Start with 2,4,6 and remove 2.
4 and 6.
Find P(odd or a 6) for a fair die.
2 marks
Favourable results are 1,3,5,6: four of six.
$2 / 3$
One person is chosen at random from the displayed group. Use the diagram to find P(A and B).
2 marks
The overlap is 4 of 20.
$1 / 5$
One person is chosen at random from the displayed group. Use the diagram to find P(A or B).
2 marks
$6 + 4 + 3 = 13$ people are in at least one group.
$13 / 20$
Of 30 students, 18 play football, 12 tennis and 5 both. How many play at least one?
2 marks
$18 + 12 - 5 = 25$ ; subtract the double-counted overlap.
25
Of 30 students, 18 play football, 12 play tennis and 5 play both. One student is chosen at random. Find P(neither sport).
2 marks
$30 - 25 = 5$ play neither.
$5 / 30 = 1 / 6$ .
$1 / 6$

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Year 9 compound-event sample spaces, probabilities, experiments and simulations.

Official curriculum source ↗

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