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§ Probability

Systematic Listing

§ Probability

Systematic Listing

CCSS.7.SP3 min read

Systematic listing is a method for finding all possible outcomes of a probability experiment by organising them in a structured way using tables, tree diagrams, or ordered lists. This technique ensures no outcomes are missed or counted twice when calculating probabilities. The method appears in Year 8 of the UK National Curriculum as students begin working with more complex probability scenarios.

§ 01

Why it matters

Systematic listing forms the foundation for calculating probabilities in real-world scenarios where multiple events occur together. When planning a school timetable with 5 subjects and 6 time slots, systematic listing reveals all 30 possible combinations. Sports tournaments use this method to determine fixture lists — a 4-team round-robin tournament requires listing 6 unique match pairings. The technique becomes essential for GCSE probability questions involving compound events like drawing cards or rolling multiple dice. Quality control in manufacturing relies on systematic listing to identify all possible defect combinations across product lines. Without this organised approach, calculating accurate probabilities for insurance premiums, weather forecasting, or medical diagnosis would be impossible, as missing even one outcome skews the entire probability calculation.

§ 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 8-sided die.

Answer: 1, 2, 3, 4, 5, 6, 7, 8

  1. List each face → 1, 2, 3, 4, 5, 6, 7, 8 — A 8-sided die has faces numbered 1 to 8.
  2. Count → 8 outcomes — There are 8 possible outcomes.
Medium§ 03

List all outcomes of flipping 3 coins.

Answer: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT

  1. Systematically list: coin 1 × coin 2 × coin 3 → HHH, HHT, HTH, HTT, THH, THT, TTH, TTT — For each outcome of the first event, list all outcomes of the second (and third, if any).
  2. Count → 8 outcomes (2 × 2 × 2 = 8) — The total is the product of individual outcome counts.
§ 04

Common mistakes

  • When listing outcomes for 2 coins, writing only 3 possibilities (HH, HT, TT) instead of the correct 4 outcomes (HH, HT, TH, TT), missing that HT and TH are different arrangements.
  • For rolling 2 dice with sum of 7, counting (3,4) and (4,3) as the same outcome instead of recognising them as 2 separate outcomes from the 6 total ways to make 7.
  • Listing outcomes for 3 coins as 6 possibilities instead of the correct 8, forgetting to apply the multiplication principle (2 × 2 × 2 = 8).
§ 05

Frequently asked questions

What is the difference between systematic listing and tree diagrams?
Systematic listing is the general method of organising all outcomes, whilst tree diagrams are one specific tool for systematic listing. Tree diagrams work best for sequential events like flipping multiple coins, showing branches for each stage. Tables and ordered lists are alternative systematic approaches better suited for simultaneous events.
How do you count outcomes when events happen at the same time?
Use the multiplication principle: multiply the number of outcomes for each individual event. Rolling 2 dice simultaneously gives 6 × 6 = 36 total outcomes. For 3 coins flipped together, calculate 2 × 2 × 2 = 8 total outcomes, then list them systematically.
Why does systematic listing matter for probability calculations?
Probability equals favourable outcomes divided by total outcomes. Without systematic listing, determining either number accurately becomes impossible. Missing outcomes leads to incorrect probability calculations, whilst double-counting inflates the total, making the final probability wrong.
When should you use tables versus tree diagrams for listing?
Use tree diagrams for sequential events where order matters, like drawing cards without replacement. Use tables for simultaneous events or when comparing two variables, such as recording all possible sums when rolling 2 dice. Both methods produce identical outcome lists.
How do you check if your systematic listing is complete?
Apply the multiplication principle to verify your total count matches the expected number of outcomes. For independent events, multiply individual outcome counts together. Ensure each outcome appears exactly once in your list and follows your chosen organisational pattern consistently.
§ 06

See also

ProbabilityMultiple
§ 06

Where to next?

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