A sample space is the set of possible outcomes of a specified experiment or random process. An event is a collection of outcomes within that space. Define the process carefully before listing outcomes or assigning probabilities.
Specify what is observed
A process such as rolling a die can be described by the face shown, while rolling twice can be described by an ordered pair. The sample space depends on what the experiment records. Do not switch from ordered results to an unordered total halfway through a calculation without explaining the change.
List outcomes without double counting
For two recorded coin tosses, a simple outcome description is HH, HT, TH and TT. HT and TH are different ordered outcomes. If you record only the number of heads, the possible values are 0, 1 and 2 instead. These descriptions are linked, but their outcomes do not necessarily have equal probabilities.
Define the event clearly
An event might be “exactly one head”, which corresponds to HT or TH in the ordered description. State the event in terms of the sample space before counting. Words such as “at least”, “exactly” and “no more than” create different events. Check them against the question rather than treating them as interchangeable.
Do not assume outcomes are equally likely
Counting favourable outcomes and dividing by the total works under the appropriate equal-likelihood assumption. A sample space alone does not establish that assumption. Where probabilities differ, use the specified model or values. Explain independence separately when a calculation multiplies probabilities across events.
Check the interpretation of the answer
The result should lie between zero and one and refer to the event asked about. Explain the model and any assumptions. A probability is not a guarantee about the next individual outcome. In coursework, show enough of the sample space or event definition for the reader to follow why the calculation applies.
Sources and further reading
Related study guides
- Convenience Sampling: Meaning, Examples and Limitations
- What Is a Pilot Study? Purpose, Feasibility and Reporting
- Theoretical Framework: Meaning and How to Build One
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