Outcomes and sample spaces | Year 7

01 / LEARN

Understand the idea.

Trial, outcome and sample space

A trial is one performance of a chance action. An outcome is the result you record. Decide the recording rule first: when rolling a die, record its upper face number, rather than its position on the table.

The sample space lists every possible recorded outcome, once each. For an ordinary die, $S = {1, 2, 3, 4, 5, 6}$ . Check your list against the device: missing a face makes the list incomplete, while repeating a number adds no new outcome.

Events are groups of outcomes

An event is a collection of outcomes satisfying a condition. Start with the full sample space, then test each member. For a die result greater than $4$ , keep $5$ and $6$ ; exclude $4$ because the condition says greater than, not equal to.

An impossible event has no outcomes in the sample space; a certain event includes them all. Listing possible outcomes says what can happen. It does not say that every listed outcome has the same chance.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Choose a numbered card

A card is chosen from six cards labelled 1, 2, 3, 4, 5 and 6. List the sample space and the event “an even number”.

Worked solution

1
List every number that could be selected.
$S = {1, 2, 3, 4, 5, 6}$
Each of the six card labels is a possible recorded result.
2
Identify the even labels.
$\begin{aligned} 2 & = 2 \times 1 \\ 4 & = 2 \times 2 \\ 6 & = 2 \times 3 \end{aligned}$
An even whole number is divisible by two without a remainder.
3
Write the event separately from the sample space.
$E = {2, 4, 6}$
Only these three members of the sample space satisfy the even-number condition.

Answer

$S = {1, 2, 3, 4, 5, 6}$ ; even event $= {2, 4, 6}$ .

EXAMPLE 02

A careful condition

An ordinary die is rolled once. List the outcomes for “a factor of 6”.

Worked solution

1
Begin with the possible die results.
$S = {1, 2, 3, 4, 5, 6}$
The event can contain only numbers that the die can show.
2
Find factor pairs of six.
$\begin{aligned} 6 & = 1 \times 6 \\ 6 & = 2 \times 3 \end{aligned}$
Each number in these pairs divides six exactly; four and five do not.
3
Collect the factor outcomes without repeats.
$E = {1, 2, 3, 6}$
Both one and six are factors, so neither should be omitted.

Answer

${1, 2, 3, 6}$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

A coin is tossed once. What is its ordinary two-outcome sample space?

Show answer & reasoning

${H, T}$ , where H is heads and T is tails.

The ordinary recording rule distinguishes heads from tails, so $S = {H, T}$ . These are the two recorded outcomes; neither is left out because you prefer the other.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

List the sample space for one ordinary six-sided die roll.
2 marks
One card is selected from cards labelled M, A, T, H and S. List the sample space.
2 marks
For a die roll, list the outcomes in the event “prime number”.
2 marks
For a die roll, list the outcomes “less than 4”.
2 marks
Is “roll an 8” impossible, possible or certain on an ordinary die? Explain.
2 marks
A die is rolled. Is the event “less than 7” certain? Explain.
2 marks

Read the worked solutions

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

List the sample space for one ordinary six-sided die roll.
2 marks
One face number is recorded.
Include each of the six face numbers once.
${1, 2, 3, 4, 5, 6}$ .
One card is selected from cards labelled M, A, T, H and S. List the sample space.
2 marks
The recorded outcome is the selected letter.
All five distinct letters belong in the list.
${M, A, T, H, S}$ .
For a die roll, list the outcomes in the event “prime number”.
2 marks
Prime numbers have exactly two positive factors.
The primes among 1 to 6 are 2, 3 and 5; 1 is not prime.
${2, 3, 5}$ .
For a die roll, list the outcomes “less than 4”.
2 marks
Check each possible face against the condition.
4 is not less than 4.
${1, 2, 3}$ .
Is “roll an 8” impossible, possible or certain on an ordinary die? Explain.
2 marks
8 is not in the sample space 1 to 6.
No allowed outcome meets the condition.
Impossible.
A die is rolled. Is the event “less than 7” certain? Explain.
2 marks
All six results 1, 2, 3, 4, 5 and 6 are less than 7.
The event includes every possible result.
Yes.

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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 ↗

In this lesson

Understand the idea.

Trial, outcome and sample space

Events are groups of outcomes

See the method in action.

Choose a numbered card

List every number that could be selected.

Identify the even labels.

Write the event separately from the sample space.

A careful condition

Begin with the possible die results.

Find factor pairs of six.

Collect the factor outcomes without repeats.

Try it yourself.

Check your understanding

Keep practising