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Listing Outcomes

FoundationChallengeAQAEdexcelOCR

Practise listing outcomes with this free Foundation GCSE Maths worksheet from Virtus Academy. You'll work through systematically listing outcomes, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. List outcomes systematically so you don't miss or repeat any.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

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Topic overview

Listing outcomes systematically ensures none is missed when finding a probability. Working randomly almost always leaves gaps.

A sample space diagram is a table showing every combination of two events. For two dice, a six-by-six grid gives all \(36\) outcomes, and counting those meeting a condition gives the probability directly.

For combinations without a natural grid, list them in a fixed order — alphabetical, or numerical, or by holding the first item constant while varying the second. Any consistent system works; the important thing is that it guarantees completeness.

Revision notes

Sample space diagrams

Draw a table with one event's outcomes along the top and the other's down the side.

Each cell gives a combined outcome. For two dice showing totals, the grid contains \(36\) cells, and counting those equal to \(7\) gives \(6\), so the probability is \(\frac{6}{36}\).

Systematic listing

Hold the first item fixed and vary the second, then move on.

For pairs from A, B, C: AB, AC, then BC. That gives three pairs, and the order guarantees none is repeated or missed.

Counting the outcomes

Once listed, count how many meet the condition and divide by the total.

The systematic approach is what makes the total reliable, which is why exam questions award marks for the listing itself.

Key points

  • List outcomes systematically to avoid missing any.
  • A sample space diagram shows all combinations.
  • Two dice give 36 outcomes.
  • Hold one item fixed and vary the other.
  • Count favourable outcomes over the total.
  • Marks are awarded for the systematic listing.

Worked examples

Example 1

Two coins are flipped. List all the outcomes.

Working

\[\text{Hold the first coin fixed}\]use a systematic order
\[HH, HT, TH, TT\]list all four outcomes

Example 2

Two dice are rolled. How many outcomes are there in total?

Working

\[6 \times 6\]each die has six outcomes
\[= 36\]state the total

Example 3

Two dice are rolled. Find the probability the total is \(7\).

Working

\[\text{Six combinations give } 7\]1+6, 2+5, 3+4, 4+3, 5+2, 6+1
\[\frac{6}{36} = \frac{1}{6}\]favourable over total, simplified

Common mistakes

  • Listing randomly.

    A system is what guarantees completeness; random listing misses outcomes.

  • Forgetting that order can matter.

    HT and TH are different outcomes when flipping two coins in sequence.

  • Miscounting the total.

    For two dice it is 6 × 6 = 36, not 12.

  • Not showing the listing.

    Marks are awarded for the systematic list, not just the final probability.

Exam tips

  • Use a sample space diagram whenever there are two events.
  • Hold one item fixed and vary the other.
  • Count the total outcomes carefully.
  • Show your list or grid in the working.

Key terms

Sample space
The set of all possible outcomes.
Systematic
Following a fixed order to guarantee completeness.
Outcome
One possible result.
Combination
A pairing of results from two events.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.