Sample Space and Events

All possible outcomes, and the subsets we actually care about.

Before we can compute any probability, we need to be precise about what we're modeling. The sample space, written $\Omega$ , is the set of all possible outcomes of an experiment. An event is a subset of $\Omega$ — the specific outcomes we care about.

For a single die roll, the sample space is

\Omega = \{1, 2, 3, 4, 5, 6\}

The event "rolled an even number" is the subset $E = \{2, 4, 6\}$ . When all outcomes are equally likely, the probability of an event is just the size of the event divided by the size of the sample space:

P(E) = \frac{|E|}{|\Omega|} = \frac{3}{6} = \frac{1}{2}

Events combine using set operations. The union $A \cup B$ means " $A$ or $B$ or both." The intersection $A \cap B$ means "both $A$ and $B$ ." The complement $A^c$ means "not $A$ ." These set operations translate directly into probability rules — for instance, $P(A^c) = 1 - P(A)$ — which is why precise sample-space thinking pays off on hard problems.