Theory

Theory 1

Events and outcomes – informally

• An event is a description of something that can happen.
• An outcome is a complete description of something that can happen.

All outcomes are events. An event is usually a partial description. Outcomes are events given with a complete description.

Here ‘complete’ and ‘partial’ are within the context of the probability model.

It can be misleading to say that an ‘outcome’ is an ‘observation’.

• ‘Observations’ occur in the real world, while ‘outcomes’ occur in the model.
• To the extent the model is a good one, and the observation conveys complete information, we can say ‘outcome’ for the observation.

Notice: Because outcomes are complete, no two distinct outcomes could actually happen in a run of the experiment being modeled.

When an event happens, the fact that it has happened constitutes information.

Events and outcomes – mathematically

• The sample space is the set of possible outcomes, so it is the set of the complete descriptions of everything that can happen.
• An event is a subset of the sample space, so it is a collection of outcomes.

For mathematicians: some “wild” subsets are not valid events. Problems with infinity and the continuum...

Notation

• Write $S$ for the set of possible outcomes, $s\in S$ for a single outcome in $S$.

• Write $A,B,C,\cdots \subset S$ or $A_1,A_2,A_3,\cdots \subset S$ for some events, subsets of $S$.

• Write $ℱ$ for the collection of all events. This is frequently a huge set!

• Write $|A|$ for the cardinality or size of a set $A$, i.e. the number of elements it contains.

Using this notation, we can consider an outcome itself as an event by considering the “singleton” subset $\{\omega \}\subset S$ which contains that outcome alone.

Theory 2

New events from old

Given two events $A$ and $B$, we can form new events using set operations:

$$\begin{array}{cc}A\cup B⟷& \text{“event }A\text{ OR }\text{event }B\text{”}\\ & \\ A\cap B⟷& \text{“event }A\text{ AND }\text{event }B\text{”}\\ & \\ A^c⟷& \text{𝐧𝐨𝐭}\text{ event }A\end{array}$$

We also use these terms for events $A$ and $B$:

• They are mutually exclusive when $A\cap B=\varnothing$, that is, they have no elements in common.

• They are collectively exhaustive $A\cup B=S$, that is, when they jointly cover all possible outcomes.

In probability texts, sometimes $A\cap B$ is written “$A\cdot B$” or even (frequently!) “$AB$”.

Rules for sets

Algebraic rules

• Associativity: $(A\cup B)\cup C=A\cup (B\cup C)$. Analogous to $(A+B)+C=A+(B+C)$.

• Distributivity: $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$. Analogous to $A(B+C)=AB+AC$.

De Morgan’s Laws

• ${(A\cup B)}^c=A^c\cap B^c$

• ${(A\cap B)}^c=A^c\cup B^c$ In other words: you can distribute “ ${}^c$ ” but must simultaneously do a switch $\cap \leftrightarrow \cup$.