Sample Spaces and Events
A sample space lists every possible outcome of an experiment; an event is any subset of those outcomes.
The building blocks
Probability starts with an experiment whose result is uncertain. The sample space S is the set of all possible outcomes. A single roll of a six-sided die has S = {1,2,3,4,5,6}. A measurement of plasma temperature that could take any positive real value has an uncountably infinite sample space.
Events as subsets
An event is any subset of S. Rolling an even number is the event {2,4,6}. The event that always happens is S itself; the event that never happens is the empty set. Because events are sets, the operations of set theory apply directly.
- Union A ∪ B: A or B (or both) occurs.
- Intersection A ∩ B: both A and B occur.
- Complement Aᶜ: A does not occur.
- Disjoint (mutually exclusive) events: A ∩ B = ∅.
From sets to numbers
A probability measure assigns a number in [0,1] to each admissible event. The structure of the sample space decides what kind of measure is possible: a finite or countable S supports a probability mass on individual outcomes, while a continuous S needs a density and a sigma-algebra of measurable events to avoid paradoxes.
Why the framing matters in practice
In simulation work for a machine like the breeder Hyperion, the sample space might be the set of plausible input configurations for a Monte Carlo neutronics run, and an event might be 'the wall heat flux exceeds a threshold'. Defining S precisely, before assigning any probabilities, prevents double-counting and undefined outcomes later.
A well-posed sample space is exhaustive (covers every outcome) and its elementary outcomes are mutually exclusive. Getting this right is the quiet prerequisite for every formula that follows.