Event (Probability Theory)

probability-theory

Definition

Event

An event is a subset of the sample space, i.e. $A\subseteq \Omega$, i.e. the set of desired elementary events.

Examples:

• $\varnothing \widehat{=}\text{impossible event}$
• $\Omega \widehat{=}\text{certain event}$
• $A^c\widehat{=}\text{opposite of event}A$ (complement)

Events are given by pre-images $X^{-1}(\mathcal{X})=\{\omega \in \Omega \mid X(\Omega )\in \mathcal{X}\}$ for $\mathcal{X}\subseteq \mathbb{R}$, which is how probabilities are assigned.

Set of all Events

Definition

Set of All Events

Let $\Omega$ be a sample space. Its set of all admissible events is a family

$$\mathfrak{S}\subseteq \mathcal{P}(\Omega )\cong 2^{\Omega }$$

that forms a sigma-algebra on $\Omega$. Each $A\in \mathfrak{S}$ is an event to which a probability may be assigned. Together with a probability measure $P:\mathfrak{S}\to [0,1]$, it forms the probability space $(\Omega ,\mathfrak{S},P)$.

Link to original

Independence

Definition

Independent Event

An event $A$ is called independent from event $B$ if:

$$P(A,B)\equiv P(A\cap B)=P(A)\cdot P(B)$$

It’s denoted as $A\perp B$ or $A\perp \perp B$.

A set of events is called independent if they are pair-wise independent.

Link to original