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List outcomes in a sample space

A sample space is the complete set of distinct possible outcomes of a finite random experiment, with each outcome represented precisely and without omission or duplication. The learner systematically organizes outcomes for simple one- or two-stage experiments using lists, tables, or tree diagrams, distinguishing individual outcomes from events and recognizing when order matters; advanced counting formulas, permutations and combinations, and infinite or continuous sample spaces are not included.

Detailed Explanation: List outcomes in a sample space

A sample space is a list of every possible outcome of a random experiment. Each outcome should be:

  • included exactly once,
  • written clearly, and
  • different from every other outcome.

For a two-step experiment, keep the outcomes in the order the steps happen.

Example: A coin is flipped, and then a spinner with the numbers (1,2,3)(1,2,3) is spun. List the sample space.

Step 1: List the results of the first step.

The coin can land:

H or TH \text{ or } T

Step 2: For each coin result, pair it with every spinner result.

If the coin is HH, the outcomes are:

(H,1), (H,2), (H,3)(H,1),\ (H,2),\ (H,3)

If the coin is TT, the outcomes are:

(T,1), (T,2), (T,3)(T,1),\ (T,2),\ (T,3)

Step 3: Write the complete sample space.

S={(H,1),(H,2),(H,3),(T,1),(T,2),(T,3)}S=\{(H,1),(H,2),(H,3),(T,1),(T,2),(T,3)\}

There are 66 possible outcomes. The pair ((H,2))((H,2)), for example, means the coin landed heads and the spinner landed on 22.

The order matters because the first part records the coin and the second part records the spinner. The sample space lists individual outcomes; an event would be a group of outcomes, such as “the spinner lands on 22”:

{(H,2),(T,2)}\{(H,2),(T,2)\}

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    Probability Sample Space - Definition to Sample Space List

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