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§ Probability

Systematic Listing

CCSS.7.SP3 min read

Year 8 students often struggle to find all possible outcomes when dealing with compound probability events. Systematic listing provides the foundation for accurate probability calculations by ensuring no outcomes are missed or double-counted.

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§ 01

Why it matters

Systematic listing forms the backbone of probability theory in GCSE mathematics, appearing in both Foundation and Higher tier papers. Students use these skills when calculating lottery odds (14 million combinations), analysing sports fixtures (380 Premier League matches from 20 teams), or determining password security (268 combinations for 8-letter passwords). The method directly supports the Year 8 National Curriculum requirement to enumerate sets systematically. Beyond exams, this logical approach helps students organise complex information in science experiments, where listing all possible variable combinations ensures thorough investigation. Restaurant managers use systematic listing to create shift rotas covering 7 days with 15 staff members, generating 105 possible combinations. The skill transfers to computer science, where algorithms require exhaustive searches through all possible states to find optimal solutions.

§ 02

How to solve systematic listing

Systematic Listing

  • List all possible outcomes in an organised way.
  • Use a table, tree diagram, or ordered list.
  • Count the total number of outcomes.
  • Use the list to find probabilities.

Example: Two dice: list all 36 pairs from (1,1) to (6,6).

§ 03

Worked examples

Beginner§ 01

List all outcomes of flipping a coin.

Answer: H, T

  1. Identify possible outcomes → Heads (H), Tails (T) — A coin has two sides.
  2. Write the sample space → S = {H, T} — 2 possible outcomes.
Easy§ 02

List all outcomes of rolling a 6-sided die.

Answer: 1, 2, 3, 4, 5, 6

  1. List each face → 1, 2, 3, 4, 5, 6 — A 6-sided die has faces numbered 1 to 6.
  2. Count → 6 outcomes — There are 6 possible outcomes.
Medium§ 03

List all outcomes of flipping a coin and rolling a 6-sided die.

Answer: H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6

  1. Systematically list: coin × die face → H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6 — For each outcome of the first event, list all outcomes of the second (and third, if any).
  2. Count → 12 outcomes (2 × 6 = 12) — The total is the product of individual outcome counts.
§ 04

Common mistakes

  • Students list outcomes randomly without structure, missing pairs like (3,5) whilst including (5,3) twice when rolling two dice, getting 32 outcomes instead of 36.
  • Pupils confuse order significance, treating coin-then-die as identical to die-then-coin, listing HT and TH as the same outcome instead of recognising 4 distinct results.
  • Learners forget to multiply outcome counts, adding 2 + 6 = 8 for coin-and-die combinations rather than calculating 2 × 6 = 12 total possibilities.
  • Students create incomplete tree diagrams, drawing only 3 branches from each coin flip instead of 6 when combining with a die roll.
Practice on your own
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§ 05

Frequently asked questions

When should students use tables versus tree diagrams for systematic listing?
Use tables for independent events like rolling two dice (36 outcomes in a 6×6 grid). Choose tree diagrams for sequential events like drawing cards without replacement, where second outcomes depend on first results. Tables work better when both events have similar outcome counts.
How do I help students remember to list outcomes systematically rather than randomly?
Teach the 'fix-first-vary-second' method. Fix the first coin as Heads, list all die outcomes H1-H6, then fix Tails and repeat T1-T6. This prevents missing combinations and reduces counting errors significantly.
Should students always write out every single outcome for large sample spaces?
For GCSE purposes, list outcomes when totals are under 20. For larger spaces like 52-card combinations, teach students to count systematically using multiplication principles whilst showing the first few examples to demonstrate understanding.
What's the difference between systematic listing and using the multiplication principle?
Systematic listing shows every individual outcome (H1, H2, H3, etc.) whilst multiplication calculates total count (2 × 6 = 12). Use listing for finding specific probabilities, multiplication for counting large sample spaces efficiently.
How do I mark systematic listing questions when students miss just one outcome?
Award method marks for systematic approach even with minor omissions. Give full marks for complete accurate lists. Deduct minimally for single missing outcomes if the method shows clear understanding of systematic organisation principles.
§ 06

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