What Is a Sample Space?
When we conduct an experiment — like flipping a coin, rolling a die, or picking a card — there are a number of possible results. The complete collection of every possible result is called the sample space.
The sample space of an experiment is the set of all possible outcomes. It is often denoted by the letter or the Greek letter .
For a single coin toss, the sample space is:
For rolling a standard six-sided die:
The total number of outcomes in the sample space is important because it forms the denominator when we calculate probabilities.
Think of the sample space like a menu at a restaurant — it lists every single option available to you. You can only order what's on the menu, and probability tells you how likely you are to pick each dish if you chose randomly.
Listing Sample Spaces for Combined Events
Things get more interesting when we combine two or more experiments. For example, what if you flip a coin and roll a die? Now you need to list every possible combination of outcomes.
A combined event (or compound event) involves two or more simple experiments performed together or in sequence.
For a coin flip and a die roll, each of the 2 coin outcomes can pair with each of the 6 die outcomes, giving:
The full sample space is:
To find the total number of outcomes for combined independent events, multiply the number of outcomes for each individual event. This is called the counting principle (or multiplication principle).
We can organize these combinations using tables, lists, or — most powerfully — tree diagrams.
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