Probability language and sample spaces

• An experiment is a repeatable chance process, such as rolling a die.
• An outcome is one result, such as rolling 5.
• The sample space is the set of every possible outcome. For a die it is $\{1,2,3,4,5,6\}$.
• An event is a set of outcomes. “Odd” is the event $\{1,3,5\}$.

This counting rule is valid only when the elementary outcomes have equal probability. A fair die has equally likely faces. A random selection from a stated group gives each item an equal chance, but grouped results such as two-dice totals may still have different probabilities.

1. List or represent every elementary outcome. Use a set, table, grid or tree so that no possible result is missed.
2. Check equal likelihood. Only then may you count successful outcomes over all outcomes.
3. Count and simplify. The numerator counts outcomes in the event; the denominator counts the whole sample space.

Enumeration means listing every elementary outcome. For a large unordered selection, combinations count the equally likely groups instead; the lesson “Combinations in probability” develops that route fully.

$A'$ means “not A”. The event and its complement cover the whole sample space without overlap, so their probabilities add to 1.

• “not”, “none” and “fails” may directly describe a complement.
• Repeated observations estimate an unknown probability with relative frequency: $\text{relative frequency}=\frac{\text{successes}}{\text{trials}}$. It is an estimate, not a guaranteed exact value.
• Over $n$ future trials, the expected number of successes is $nP(A)$. “Expected” means a long-run average, not a promise for one run.