EVENT

Tags: #◇Computer Science #◇STT101A #◇Conceptual #◇Event

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◇Overview

In probability theory, an Event is any subset $E$ of the sample space $S$ ($E \subseteq S$) to which a probability is assigned. An event occurs on a given trial of a statistical experiment if and only if the observed elementary outcome $\omega \in S$ belongs to $E$ ($\omega \in E$).

◇Key Definitions

• Simple (Elementary) Event: An event containing exactly one sample point, $E = \{\omega\}$ with $|E| = 1$.
• Compound Event: An event containing two or more sample points ($|E| \ge 2$), expressible as the union of simple events.
• Sure (Certain) Event ($S$): The entire sample space $S$, which always occurs ($P(S) = 1$).
• Impossible (Null) Event ($\emptyset$): The empty subset $\emptyset$, which contains no outcomes and can never occur ($P(\emptyset) = 0$).
• Event Space ($\mathcal{F}$): The $\sigma$-algebra of subsets of $S$ (for finite $S$, the power set $\mathcal{P}(S)$ of size $2^{|S|}$).

◇Worked Example

Problem: A coin is tossed twice, giving $S = \{HH, HT, TH, TT\}$. Classify the following events and state their cardinalities:

1. $E_1 =$ "both tosses are heads" $= \{HH\}$ ($|E_1| = 1$, simple event).
2. $E_2 =$ "at least one head appears" $= \{HH, HT, TH\}$ ($|E_2| = 3$, compound event).
3. $E_3 =$ "three heads appear in two tosses" $= \emptyset$ ($|E_3| = 0$, impossible event).

◇References & Navigation

• Previous Topic: ◇Sample Space
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• Related Notes: ◇Statistical Experiment, ◇Union