**What topics are covered in this presentation about basic probability?**
The topics covered in this presentation about basic probability are:
* Set theory.
* Elements of probability.
* Conditional probability.
* Sequential calculation of probability.
* Total probability and Bayes Rule.
* Independence.
* Counting.
# Set Theory Basics
**What is a set?**
A set is a collection of objects.
> **What are the objects of a set called?**
> The objects of a set are called *elements.*
>
> **How do you say an element $A$ is in a set $\omega$?**
> To say an element $A$ is in a set $\omega$, you write:
> $A\in\omega$
>
> **What is a set with no elements called?**
> A set with no elements is called *the empty set.*
> > **What symbol represents the empty set?**
> > The symbol which represents the empty set is *$\emptyset$.*
**What are the three different types of sets?**
The three different types of sets are:
1. Finite - $A=\{\omega_1,\omega_2,\dots,\omega_n\}$
2. Countably infinite - $A=\{\omega_1,\omega_2,\dots\}$
3. Uncountable - A set which takes a continuous set of values like the interval $[0,1]$, the real line, etc.
**What is the other notation which can be used to describe a set?**
The other notation which can be used to describe a set is *set builder notation.*
> [!example] Example of set builder notation for describing a set $A$ containing the interval $[0,1]$
> $A=\{w:0\leq\omega\leq1\}$
**What does $B\subset A$ mean if both $A$ and $B$ are sets?**
If $A$ and $B$ are sets, $B\subset A$ means that *all of the elements in the set $B$ are in the set $A$.*
> **What is a set $B$ called if all of the elements in it are also in a set $A$?**
> If all of the elements in a set $B$ are also in a set $A$, then B is called *a subset of set $A$.*
**What is the universal set?**
The universal set is *a set containing all of the elements under discussion, like the sample space for a random experiment.*
> **What symbol represents the universal set?**
> The symbol which represents the universal set is *$\Omega$.*
**What is complementation?**
Complementation is *creating the complement of a set with respect to another set.*
> **What is the complement of a set $A$ with respect to a set $B$?**
> The complement of a set $A$ is *the set of all elements which aren't in set $A$ but exist within set $B$.*
>
> **How do you describe the complement of a set $A$ with respect to a set $B$ in set builder notation?**
> To describe the complement of a set $A$ with respect to a set $B$ in set builder notation, you write:
> $A^c=\{\omega\in B:\omega\notin A\}$
>
> **What is the complement of the universal set?**
> The complement of the universal set is *the empty set.*
**What is an intersection?**
An intersection is *the set of all elements which are in both sets.*
> **How do you describe the intersection of a set $A$ and a set $B$ in set builder notation?**
> To describe the intersection of a set $A$ and a set $B$ in set builder notation, you write:
> $A\cap B=\{\omega:\omega\in A\textrm{ and }\omega\in B\}$
**What is a union?**
A union is *the set of all elements that are in either set.*
> **How do you describe the union of a set $A$ and a set $B$ in set builder notation?**
> To describe the union of a set $A$ and a set $B$ in set builder notation, you write:
> $A\cup B=\{\omega:\omega\in A\textrm{ or }\omega\in B\}$
...
**What does it mean for a set $A$ and a set $B$ to be disjoint or mutually exclusive?**
A set $A$ and a set $B$ are disjoint or mutually exclusive *when they don't have any elements in common.*
> **What is the intersection of disjoint sets equal to and how is this expressed?**
> The intersection of disjoint set is equal to *the empty set and it's expressed as:*
> $A\cap B=\emptyset$
**What does it mean for a collection of sets $A_1,A_2\dots,A_n$ to partition a set $B$ and how is this expressed?**
A collection of sets $A_1,A_2\dots,A_n$ partitions a set $B$ *when they're disjoint and the union of all of them equals set $B$, and it's expressed as:*
$\bigcup\limits_{i=1}^nA_i=B$
...
> [!example] Examples of Venn Diagram representations for sets, complements, intersections, and unions
> ![[Venn Diagram Representations For Sets, Complements, Intersections, and Unions.png]]
>
...