Outcomes and sample spaces | Year 7

01 / LEARN

The idea, step by step

Trial, outcome and sample space

A trial is one performance of a chance action, such as rolling one die. The outcome is the result recorded. Decide what you will record before you start: for a die this is usually the number on the upper face, not how far the die travelled.

The sample space lists every result that could be recorded. Use braces to show a collection, and do not repeat the same recorded result. An ordinary six-sided die has sample space $S = {1, 2, 3, 4, 5, 6}$ . A complete list helps prevent overlooked outcomes.

Events are groups of outcomes

An event is a condition that some outcomes satisfy. For “greater than 4”, the die outcomes are ${5, 6}$ . Write the full sample space first, then select the outcomes meeting the condition.

An impossible event contains no outcomes from the sample space. A certain event contains them all. Listing outcomes does not prove they have equal chances; that is an extra assumption about the device or selection method.

A common trap

Writing only the desired results as the sample space.

Instead: The sample space includes all possible recorded outcomes; the desired event is a subset of it.

02 / WORKED EXAMPLES

See the thinking

Example 1

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”.

Possible recorded numbers are 1, 2, 3, 4, 5 and 6.
Even numbers are divisible by 2: select 2, 4 and 6.
Keep the event separate from the full sample space.

S = {1, 2, 3, 4, 5, 6}

; even event

= {2, 4, 6}

Example 2

A careful condition

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

A factor divides 6 with no remainder.
Check every die outcome: 1, 2, 3 and 6 divide 6 exactly.
Include 6 itself; a factor can equal the original number.

{1, 2, 3, 6}

Check your understanding

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

Show answer

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

The recorded upper face can be either heads or tails.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF 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 ↗

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