What Is a Trial and an Outcome?
Trial: A trial is a single execution of a random experiment , one run of the process whose result we want to observe.
Outcome: An outcome is a specific result of a single trial. It is one of the possible things that can happen when you run the experiment.
Every time you run a random experiment, you are performing a trial, and whatever happens is an outcome. The key word is random , you cannot predict with certainty which outcome will occur before the trial takes place.
Common trials and their outcomes:
- Flipping a coin → outcome is Heads or Tails
- Rolling a six-sided die → outcome is one of 1, 2, 3, 4, 5, 6
- Drawing a card from a standard deck → outcome is one of 52 possible cards
- A student taking a multiple-choice question → outcome is the answer chosen (e.g. A, B, C, or D)
A trial must be repeatable under the same conditions. Each repetition is an independent trial, and the collection of results over many trials is what lets us study probability experimentally.
The Sample Space
Sample Space (U): The sample space, denoted U, is the set of all possible outcomes of an experiment. It lists every result that could occur in a single trial.
The sample space is the foundation for all probability calculations , you need to know every possible outcome before you can work out how likely any particular result is.
Common examples:
- Single coin flip:
- Single die roll:
- Flipping two coins:
Ways to represent a sample space:
- List , simply write all outcomes inside curly brackets, e.g.
- Table , useful when two experiments are combined (e.g. rolling two dice simultaneously)
- Tree diagram , useful for sequential experiments (e.g. flipping a coin then rolling a die)
For two combined experiments, a table is usually the quickest way to list the sample space without missing any outcomes. Draw one experiment along the top and the other down the side , each cell gives you one outcome.
Rolling two dice , sample space as a table:
The table has 6 rows (die 1 values 1–6) and 6 columns (die 2 values 1–6), giving equally likely outcomes. For instance, the outcome (3, 5) means die 1 shows 3 and die 2 shows 5.