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

The set of all events $\mathcal{F}\subseteq 2^{\Omega }$ is a $\sigma$-algebra (sigma algebra). It contains $\Omega \in \mathcal{F}$ and is closed under complement, countable unions and intersections.

Independence

Definition

Independent Event

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

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

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.

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