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 for the set of possible outcomes, for a single outcome in .

• Write or for some events, subsets of .

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

• Write for the cardinality or size of a set , 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 which contains that outcome alone.

Theory 2

New events from old

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

We also use these terms for events and :

• They are mutually exclusive when , that is, they have no elements in common.

• They are collectively exhaustive , that is, when they jointly cover all possible outcomes.

In probability texts, sometimes is written “” or even (frequently!) “”.

Rules for sets

Algebraic rules

• Associativity: . Analogous to .

• Distributivity: . Analogous to .

De Morgan’s Laws

• In other words: you can distribute “ ” but must simultaneously do a switch .