In statistics and probability, a sample space is the complete set of outcomes that could happen in an experiment. An event is any chosen part of that sample space, such as rolling an even number on a die or drawing a red card from a deck. These ideas matter because every probability question starts by identifying what can happen and which outcomes count as success.
A clear outcome map helps prevent missing possibilities or counting the same outcome twice.
Simple events contain exactly one outcome, while compound events contain two or more outcomes. Lists, tables, and tree diagrams are common tools for organizing sample spaces, especially when an experiment has several stages. Once the sample space is known, probability can often be found by comparing the number of favorable outcomes to the total number of equally likely outcomes.
The basic structure is P(event) = favorable outcomes / total outcomes.
Key Facts
- Sample space S = the set of all possible outcomes of an experiment.
- An event E is a subset of the sample space, so E is contained in S.
- For equally likely outcomes, P(E) = n(E) / n(S).
- A simple event has exactly one outcome, such as rolling a 4 on a standard die.
- A compound event has more than one outcome, such as rolling an even number: {2, 4, 6}.
- For multi-step experiments, multiply choices to count outcomes: total outcomes = choices in step 1 × choices in step 2 × ...
Vocabulary
- Sample Space
- The set of all possible outcomes for a probability experiment.
- Outcome
- A single possible result of an experiment, such as heads in a coin toss.
- Event
- A set of one or more outcomes from the sample space that share a chosen condition.
- Simple Event
- An event that contains exactly one outcome.
- Tree Diagram
- A branching diagram used to list all outcomes of a multi-step experiment.
Common Mistakes to Avoid
- Leaving outcomes out of the sample space: this is wrong because probabilities depend on the complete set of possible results.
- Counting the same outcome more than once: this is wrong because repeated counting makes some outcomes seem more likely than they really are.
- Treating all outcomes as equally likely without checking: this is wrong because P(E) = n(E) / n(S) only works when each outcome has the same chance.
- Confusing an event with a single outcome: this is wrong because an event can be a subset containing many outcomes, such as all odd rolls on a die.
Practice Questions
- 1 A fair coin is flipped and a standard 6-sided die is rolled. List the sample space size and find the probability of getting heads and an even number.
- 2 A student chooses one shirt from 3 shirts and one pair of pants from 4 pairs of pants. How many total outfits are in the sample space, and how many outcomes include a specific blue shirt?
- 3 A spinner has sections labeled A, B, C, and D. Explain whether the event {A, C} is simple or compound, and describe how it fits inside the sample space.