Events

A random process (or a process modeled as random) produces distinct individual outcomes, called elementary events. We use \(E\) to denote the set of such outcomes; for a coin flip, \(E = \{H, T\}\). For a random process that produces a count, \(E = \mathbb{N}\).

Probability is defined over events. An event \(A\) is a subset of \(E\) (\(A \subseteq E\)). If elementary events are events, they are represented as singletons: \(A = \{H\}\) means “coin is heads”. \(E\), the set of all elementary events, is the event “something happened”.

We use set operations to combine events (for these examples, we consider \(E\) to be a deck of 52 standard playing cards; \(A\) is “2” and \(B\) is “red card”):

Illustration

Cards, A, and B

• \(A \cap B\) is the event “both \(A\) and \(B\) happened”; for our example, the conjunction is “red 2”, of which there are 2 (2♥, 2♦).

  Illustration of \(A \cap B\)

  

  Highlighting A∩B

• \(A \cup B\) is the event “either \(A\) or \(B\) (or both) happened”; for our example, the disjunction is “2 or red” — any 2, or any red card; this set has size 28: the 26 red cards (13 of each red suit), plus the two black 2s.

  Illustration of \(A \cup B\)

  

  Highlighting A∪B

• \(A \setminus B\) is the event “\(A\) happened but not \(B\)”. If \(B \subseteq A\), then \(A \setminus B = \emptyset\); for our example, the difference is “black 2”, because it is the set of 2s that are not red.

  Illustration of \(A \setminus B\)

  

  Highlighting A\B

With these definitions, we can now define the event space: \(\mathcal{F}\) is the set of all possible events (subsets of \(E\)). This is a set of sets. It does not necessarily contain every subset of \(E\), but it has the following properties:

• \(E \in \mathcal{F}\).
• If \(A \in \mathcal{F}\), then its complement \(A^c \in \mathcal{F}\). We say \(\mathcal{F}\) is closed under complement.
  • Since \(E \in \mathcal{F}\) and \(E^c = \emptyset\), \(\emptyset \in \mathcal{F}\).
• If \(A_1, A_2, \dots, A_n \in \mathcal{F}\), then their union \(\bigcup_i A_i \in \mathcal{F}\). This applies also to unions of countably many sets. We say \(\mathcal{F}\) is closed under countable unions.

\(\mathcal{F}\) is called a sigma algebra (or sigma field). For a finite set \(E\), we usually use \(\mathcal{F}= \mathcal{P}(E)\), the power set of \(E\). This means that every possible subset of \(E\) (and therefore every conceivable set of elementary events) is an event.

Here are some additional properties of sigma algebras (these are listed separately from the previous properties because those are the definition of a sigma algebra and these are consequences — we can prove them from the definitions and axioms):

• If \(A, B \in \mathcal{F}\), then \(A \cap B \in \mathcal{F}\)