What Is a Sample Space?
When you flip a coin, roll a die, or spin a spinner, you're performing an experiment — an action with uncertain outcomes. Before we can calculate any probabilities, we need to know all the possible outcomes. That's where sample spaces come in.
The sample space of an experiment is the complete set of all possible outcomes. It is often denoted by the letter or the Greek letter (omega).
A single possible result of an experiment.
For example, when you roll a standard six-sided die, the sample space is:
The sample space has 6 outcomes. Listing them all ensures we don't miss anything — and that's the entire point of defining a sample space before we start calculating probabilities.
Think of the sample space like a menu at a restaurant. Before you can figure out the chance of randomly picking a vegetarian dish, you need the complete menu first. If dishes are missing from the menu, your probability calculations will be wrong.
Listing Sample Spaces for Simple Experiments
For straightforward experiments, you can list outcomes directly using set notation (curly braces).
Flipping a coin:
Number of outcomes:
Rolling a standard die:
Number of outcomes:
Spinning a spinner with sections Red, Blue, Green, Yellow:
Number of outcomes:
Drawing a card suit from a standard deck:
Number of outcomes:
Always use to denote the total number of outcomes in a sample space. This number becomes the denominator when you calculate theoretical probabilities: .
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