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

01 / LEARN

Understand the idea.

And means overlap

And requires both conditions to hold in the same outcome. Inclusive or, the usual meaning of or in probability, requires at least one condition and includes the overlap. The phrase but not both excludes that overlap.

For a die, even results are ${2, 4, 6}$ and results greater than 3 are ${4, 5, 6}$ . Their overlap is ${4, 6}$ . Their inclusive union is ${2, 4, 5, 6}$ , with each result listed just once.

Count each region once

Read the diagram as four separate regions: 6 in A only, 4 in both, 3 in B only and 7 in neither. Inclusive or counts $6 + 4 + 3 = 13$ people; but not both counts only $6 + 3 = 9$ .

Adding the total in A to the total in B counts the overlap twice. Subtract it once to count the union correctly: $P (A or B) = P (A) + P (B) - P (A and B)$ . No assumption of independence is needed for this rule.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A die union

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

Worked solution

1
Write the outcomes for each condition.
$\begin{aligned} A & = {2, 4, 6} \\ B & = {4, 5, 6} \end{aligned}$
A records the even results, and B records the results greater than 3.
2
Combine the sets without counting repeats.
$A \cup B = {2, 4, 5, 6}$
Inclusive or keeps any result satisfying at least one condition, including 4 and 6 only once each.
3
Divide by the six equally likely die results.
$P (even or greater than 3) = \frac{4}{6} = \frac{2}{3}$
Four faces satisfy the inclusive condition on a fair die.

Answer

$4 / 6 = 2 / 3$ .

EXAMPLE 02

Exactly one group

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

Worked solution

1
Choose only the two non-overlap group regions.
$A only: 6, B only: 3$
The four people in both groups fail the phrase but not both, and the seven in neither fail both conditions.
2
Count the people in exactly one group.
$6 + 3 = 9$
A-only and B-only are separate regions, so their counts can be added.
3
Calculate the selection probability.
$P (A or B, but not both) = \frac{9}{20}$
Each of the 20 people is equally likely to be selected, and nine satisfy the event.

Answer

$9 / 20$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

Only 5.

Odd outcomes are ${1, 3, 5}$ and outcomes greater than 4 are ${5, 6}$ . Only 5 appears in both lists, so the intersection is ${5}$ .

Keep practising

6 worksheet questions · 11 marks

Open worksheet 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.
$\frac{2}{3}$
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.
$\frac{1}{5}$
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.
$\frac{13}{20}$
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.
$\frac{5}{30} = \frac{1}{6}$ .
$\frac{1}{6}$

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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 ↗

In this lesson

Understand the idea.

And means overlap

Count each region once

See the method in action.

A die union

Write the outcomes for each condition.

Combine the sets without counting repeats.

Divide by the six equally likely die results.

Exactly one group

Choose only the two non-overlap group regions.

Count the people in exactly one group.

Calculate the selection probability.

Try it yourself.

Check your understanding

Keep practising