Enumerating sample spaces and calculating probabilities

Probability • Probability

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How does ordering affect counting?

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Ordered listing treats different orders as distinct outcomes, increasing the total count when order matters.

Key concepts

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Sample space and outcomes

A sample space is the set of all possible outcomes of an experiment. Each element of the sample space is an outcome, and the total number of outcomes determines the denominator in theoretical probability calculations. When outcomes are equally likely, probability for an event equals favourable outcomes divided by total outcomes.

Equally likely outcomes

Outcomes are equally likely when each outcome has the same chance of occurring. Equal likelihood allows use of counting methods to compute exact probabilities. When outcomes are not equally likely, listing still helps but probability requires weights for each outcome.

Tables and grids for combined experiments

Tables or grids represent combinations of two experiments where each axis lists outcomes of one experiment. Filling the grid produces every ordered pair, so total outcomes equal rows times columns. Counting favourable cells then provides the probability when outcomes are equally likely.

Tree diagrams for sequential experiments

Tree diagrams show outcomes step by step for sequential experiments. Branches represent choices or events at each stage and leaves represent complete outcomes. Multiplication of branch counts gives the total number of outcomes and labelling branches with probabilities gives a visual way to compute event probabilities.

Venn diagrams and set combinations

Venn diagrams visualise relationships between sets and events such as unions, intersections and complements. Partitioning the sample space into disjoint regions prevents double counting and clarifies how to count outcomes that belong to combinations of events.

Systematic listing and avoiding errors

Systematic listing uses a chosen method (table, tree, grid, or ordered list) so that every outcome appears once. Systematic methods reduce the risk of missed or duplicated outcomes and make subsequent counting reliable for probability calculations.

Limiting factors and applicability

Enumeration methods work best when the sample space is reasonably small or when outcomes are structured. Very large or continuous sample spaces require other techniques. If outcomes are not equally likely, enumeration still helps but calculation requires probability weights instead of simple counts.

Key notes

Important points to keep in mind

List every outcome once to avoid missing or duplicating results.

Use grids for two simultaneous experiments and tree diagrams for sequential experiments.

Apply the multiplication principle: total outcomes equals product of choices at each stage.

When outcomes are equally likely, probability = favourable count ÷ total count.

Use Venn diagrams to partition the sample space and handle unions, intersections and complements.

Order outcomes explicitly when order matters; use unordered combinations when order does not matter.

If outcomes are not equally likely, assign and use individual outcome probabilities instead of simple counts.

Check counts by comparing different methods (e.g., grid vs tree) to confirm the sample space size.