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 outcome 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 simple experiments, listing the sample space is straightforward:
- Flipping a coin: — 2 outcomes
- Rolling a standard die: — 6 outcomes
- Picking a random day of the week: — 7 outcomes
Knowing the sample space is the first step in calculating probabilities, because probability is all about comparing the outcomes you care about to the total number of possible outcomes.
Sample Spaces for Combined Events
Things get more interesting when two (or more) events happen together. For example, what if you flip a coin and roll a die at the same time? You now need to list every possible combination of outcomes.
A useful way to organise this is with a sample space diagram (also called a two-way table or lattice diagram).
Flipping a coin and rolling a die
We can set up a grid:
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| H | (H,1) | (H,2) | (H,3) | (H,4) | (H,5) | (H,6) |
| T | (T,1) | (T,2) | (T,3) | (T,4) | (T,5) | (T,6) |
Total outcomes =
So the probability of getting Heads and a 3 is:
To find the total number of outcomes for combined independent events, multiply the number of outcomes of each event together. This is sometimes called the counting principle or multiplication rule.
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