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§ Probability

Systematic Listing

§ Probability

Systematic Listing

CCSS.7.SP3 min read

Systematic listing is a method for identifying and organizing all possible outcomes of a probability experiment in a structured way. This technique uses tools like tables, tree diagrams, or ordered lists to ensure no outcomes are overlooked when determining sample spaces. The method appears prominently in CCSS 7.SP as students explore compound events and calculate probabilities from complete outcome sets.

§ 01

Why it matters

Systematic listing forms the foundation for calculating accurate probabilities in real-world scenarios. Weather forecasters use systematic methods to list all possible storm paths when predicting hurricane trajectories with 95% confidence intervals. Game designers systematically catalog all 36 outcomes when rolling two dice to balance board game mechanics. Quality control engineers list all 64 possible defect combinations when testing products with 6 different components. Sports analysts systematically enumerate all 16 possible playoff bracket outcomes to calculate championship probabilities. The technique becomes essential in advanced mathematics, particularly in combinatorics where students calculate arrangements of 10 objects (3,628,800 possibilities) and in statistics courses involving complex probability distributions with hundreds of potential outcomes.

§ 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 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

  • Listing outcomes randomly produces incomplete sample spaces, such as recording only 8 outcomes for two coin flips instead of the complete set of 4: HH, HT, TH, TT.
  • Double-counting symmetric outcomes leads to incorrect totals, like counting both (2,5) and (5,2) as the same outcome when rolling two dice, reducing the sample space from 36 to 21.
  • Missing the multiplication principle results in undercounting compound events, such as listing 8 outcomes for a coin flip plus 6-sided die roll instead of the correct 12 outcomes.
§ 05

Frequently asked questions

What is the difference between systematic listing and random listing?
Systematic listing follows an organized pattern or structure to ensure all outcomes are captured, while random listing records outcomes as they come to mind. Systematic methods prevent missing outcomes and double-counting, producing complete sample spaces needed for accurate probability calculations.
How do you know if your systematic list is complete?
Multiply the number of outcomes for each individual event. For two dice, each die has 6 outcomes, so the complete list contains 6 × 6 = 36 pairs. The multiplication principle provides a check against your organized list.
When should you use a tree diagram versus a table for systematic listing?
Tree diagrams work best for sequential events with different numbers of outcomes at each stage, like drawing cards without replacement. Tables organize outcomes efficiently when all events have the same number of possibilities, such as rolling multiple identical dice.
What is a sample space in systematic listing?
A sample space is the complete set of all possible outcomes for a probability experiment. Systematic listing ensures the sample space is exhaustive and mutually exclusive, containing exactly one entry for every way the experiment can unfold.
How does systematic listing help calculate probabilities?
Systematic listing provides the denominator for probability fractions by showing the total number of equally likely outcomes. Once all 36 outcomes for two dice are listed, finding the probability of rolling a sum of 7 becomes 6/36 = 1/6.
§ 06

See also

ProbabilitySetMultipleDenominator
§ 06

Where to next?

Prerequisites
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