Experiments#
Definition: Experiment
An experiment in [[index|probability theory]] is a process with the following properties:
- The process can occur multiple times;
- All possible outcomes are known and unambiguously defined;
- It is always one and only one of the outcomes which happens.
Definition: Sample Space
The sample space of an [[Experiments#Experiments|experiment]] is the [[Sets|set]] of all possible outcomes of said experiment.
Notation
Sample spaces are often denoted by \(\Omega\).
Example: Flipping a Coin
Flipping a coin is a very simple experiment whose sample space \(S\) contains only two possible outcomes - the coin falls heads-up or the coin falls tails-up. Hence, \(S\) is just
Example: Rolling a Die
Another common experiment is the roll of a single six-sided die. There are six possible outcomes - the number on the die is 1, 2, 3, 4, 5 or 6. Hence, the sample space is
Example: Flipping Two Coins
Events#
Definition: Event
An event is any [[Sets|subset]] of the [[index#experiments|sample space]] of an [[index#experiments|experiment]].
Defined in this way, mathematical events allow us to closely model real-world conditions. However, we need a way to translate between the mathematical formalism of events and their physical reality. Hence, we say that an event has occurred if the outcome of the experiment is an element of the event.
Tip: Union of Events
The [[../Set Theory/Collections|union]] of a [[../Set Theory/Collections|collection]] of [[index#experiments|events]] is the event which occurs if and only if at least one of the events in the collection occurs.
Tip: Intersection of Events
The [[../Set Theory/Collections|intersection]] of a [[../Set Theory/Collections|collection]] of [[index#experiments|events]] is the event which occurs if and only if all of the events in the collection occur.
Tip: Complement of an Event
The [[../Set Theory/Sets]] of an event \(E\) is the event which occurs if and only if \(E\) does not occur.
Definition: Mutual Exclusiveness
A [[../Set Theory/Collections|collection]] of [[Experiments#experiments|events]] are mutually exclusive if and only if every [[../Set Theory/Sets|intersection]] between two of the [[Experiments#experiments|events]] is [[Sets|empty]].