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§ Probability·Grades 7–10

Systematic Listing Worksheets

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Easy

10 problems

Medium

20 problems

Hard

20 problems

Mixed

30 problems

Free printable systematic listing worksheets with step-by-step answer keys. Every worksheet is uniquely generated so students never see the same problems twice. Topics covered range from list outcomes of a single coin flip at the easy level through to two-dice sample space: find outcomes with a target sum at the advanced level.

CCSS.7.SPLK20.10

What is systematic listing?

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.

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.

Common mistakes to watch for

  • ✗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.

Questions teachers ask

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.
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Beginner

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Concepts
List outcomes of a single coin flip
Range
2 outcomes
Steps
1 step
Example
List all outcomes of flipping a coin

Easy

Generate →
Concepts
List outcomes of a multi-sided die
Range
6 or 8 outcomes
Steps
1–2 steps
Example
List all outcomes of rolling a 6-sided die

Medium

Generate →
Concepts
Systematically list outcomes of 2 coin flips (tree diagram)
Range
4 outcomes (2 × 2)
Steps
2 steps
Example
List all outcomes of flipping 2 coins

Hard

Generate →
Concepts
Two-dice sample space: find outcomes with a target sum
Range
36 outcomes, filter by sum
Steps
2 steps
Example
Two dice: how many outcomes give a sum of 7?

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