Sets and Probability
- (The University of Chicago - Maya Lim)
- Sets and Events
In probability theory, a set is a collection of distinct outcomes or events that are part of a larger sample space. A sample space represents all possible outcomes of a random experiment or process.
A probability event can be defined as a set of outcomes of an experiment. In other words, an event in probability is the subset of the respective sample space.
A Venn diagram is a way of visualizing sets. The universal set is represented by a rectangle and sets are represented as circles inside the universal set. Venn diagrams may prove useful when displaying a visual of how sets and probabilities are related, especially with overlapping (not mutually exclusive) events.
Here are some other things to know about sets and probability:
- A probability set function is interpretable as a probability distribution on binary sequences of fixed length.
- General probability theory deals with uncountable sets like the set of real numbers.
- The empty set, written as /0 or {}, is the set with no elements.
- Sets are usually denoted by a letter and the objects (or elements) belonging to a set are usually listed within curly brackets.
- A set is an unordered collection of objects, i.e. the order in which the elements of a set are listed does not matter.
Please refer to the following for more information:
- TAMU: Sets and Probability
- Wikipedia: Set Function
- Wikipedia: Event (Probability Theory)
- The Language of Sets
A set is a collection of items. These items are referred to as the elements or members of the set. It is useful to define the set with a symbol, usually an uppercase letter. For example, S = {a,b,c,d,e, f,g,h,i, j}.
We can say c is an element of the set {a,b,c,d,e, f,g,h,i, j} or simply write c ∈ S. The symbol ∈ is read “is an element of”. We can also say that the set R = {c} is a subset of our larger set S as every element in the set R is also in the set S.
If every element of a set A is also an element of another set B, we say that A is a subset of B and write A ⊆ B. If A is not a subset of B, we write A ⊆ B.
- Probability Set Functions
A probability set function can be defined as a function P relative to a sample space
S. The function P(C) is the probability of event C, when the following three properties hold:
P(C)≥0, for all events C
P(S)=1
A probability function P(A) measures the size of the set A. The size of the set A plus the size of the set B equals the size of the union A∪B plus the size of the intersection A∩B.
A probability is a set function that associates to each set/event a number between zero and one.
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set. The term "set function" is often used to avoid confusion between the mathematical meaning of "measure" and its common language meaning.
[More to come ...]
