Sample Space and Events — Section 1: Foundations of Probability

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.

For a single die roll, the sample space is

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

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