Systematic listing means recording every possible outcome in an organised order so that none are missed and none are repeated. It is the foundation of probability calculations at KS3 — you cannot find the probability of an event without knowing the total number of equally likely outcomes.
Why is being systematic important?
Random listing leads to missed outcomes and repeated counting. Being systematic means choosing a rule for the order — fixing one choice at a time and varying the others — so the list is both complete and non-repetitive.
Example: List all two-digit numbers made from {1, 2, 3} (digits can repeat).
Unsystematic attempt: 12, 31, 21, 13, 22, 33, 11, 23, 32. Count: 9. Correct, but easy to miss one.
Systematic attempt: Fix the tens digit and vary the units:
- Tens = 1: 11, 12, 13
- Tens = 2: 21, 22, 23
- Tens = 3: 31, 32, 33
Total: 9. The systematic list makes it obvious when the list is complete.
How do you use a sample space diagram?
A sample space diagram (or outcomes table) lists the outcomes of two events in a grid. One event's options go along the top; the other event's options go down the side. Every cell shows the combined outcome.
Worked example: Two fair coins are tossed. List all outcomes.
| Coin 2: H | Coin 2: T | |
|---|---|---|
| Coin 1: H | HH | HT |
| Coin 1: T | TH | TT |
Total outcomes: 4. P(both heads) = 1/4. P(at least one head) = 3/4.
Worked example: A four-sided spinner (1–4) and a six-sided die (1–6) are thrown. How many equally likely outcomes are there? What is the probability the total equals 7?
Total outcomes = 4 × 6 = 24. (A full table would have 24 cells.)
Outcomes giving total 7: (1,6), (2,5), (3,4), (4,3) — that is 4 outcomes.
P(total = 7) = 4/24 = 1/6
How do you list combinations of choices?
When combining choices from several categories, list all options for the first choice, then pair each with all options for the second, and so on. This method is sometimes called an ordered tree or systematic enumeration.
Worked example: A menu offers: starter (soup S or salad L), main (pasta P or chicken C), dessert (ice cream I or cake K). How many different three-course meals are possible?
Fix the starter, then branch:
- S → P → I = SPI
- S → P → K = SPK
- S → C → I = SCI
- S → C → K = SCK
- L → P → I = LPI
- L → P → K = LPK
- L → C → I = LCI
- L → C → K = LCK
Total: 8 meals. (2 × 2 × 2 = 8 confirms the count.)
How do you list arrangements (permutations)?
An arrangement (or permutation) cares about order. If you are arranging objects in a row, a different order counts as a different outcome.
Worked example: How many ways can you arrange the letters A, B, C in a row?
Fix the first letter, then list remaining options for second and third:
| First | Second | Third | Arrangement |
|---|---|---|---|
| A | B | C | ABC |
| A | C | B | ACB |
| B | A | C | BAC |
| B | C | A | BCA |
| C | A | B | CAB |
| C | B | A | CBA |
Total: 6 arrangements. (This matches 3 × 2 × 1 = 6, the pattern for 3 items in a row.)
How do you count combinations rather than arrangements?
A combination ignores order — {A, B} and {B, A} count as the same pair.
Worked example: From four students A, B, C, D, how many different pairs can be chosen for a two-person team?
List all pairs systematically (always list in alphabetical order to avoid repeats): AB, AC, AD, BC, BD, CD
Total: 6 combinations.
The link to probability: if you choose 2 students at random from 4, P(the team contains A) = number of pairs with A ÷ total pairs = 3/6 = 1/2.
Frequently asked questions
How do I know when my list is complete?
The best check is to verify the total count matches a formula. For two events with m and n outcomes each, there are m × n combined outcomes. For three events with m, n, and p outcomes, there are m × n × p. If your list count does not match, go back and look for gaps.
What is the difference between a sample space diagram and a list?
Both show all possible outcomes, but a sample space diagram is a two-way table that is easier to scan for patterns (e.g. totals that equal a target number). A list is better when you have three or more events or when the outcomes are not easily arranged in a grid.
When does order matter and when does it not?
Order matters when the items are placed in a sequence (first place, second place) or when A then B is a different outcome from B then A. Order does not matter when you are choosing a group or a subset where position is irrelevant (e.g. a team, a committee, a hand of cards).
Is systematic listing the same as the product rule for counting?
Systematic listing and the product rule give the same total, but listing shows every individual outcome while the product rule just gives the count. At KS3, listing is usually required (especially in probability) so that you can identify and count favourable outcomes. At GCSE the product rule is introduced as a shortcut for when listing everything would take too long.
For Socratic KS3 probability and counting practice with Professor Pi, see aitutors.me.