Sample Space and Probability

Sets

A set is a collection of things.
If $x$ is a thing and $S$ is a set, we write $x\in S$ if $x$ is in $S$.

Examples and Notation

• Finite: $S=\{x_1,x_2,...,x_n\}$
• Countably infinite: $S=\{x_1,x_2,...\}$
• Set based on condition: $S=\{x|x/2\in \mathbb{Z}\}$
• Uncountable: $S=\{x|0\le x\le 1\}$
• Subset: $S\subset T$
• Universal Set: $\Omega$
  • This is a set of all conceivable objects of interest in a context
• Empty set: $\varnothing$ or $\varnothing$
• Complement: $S^c$
  • ${\Omega }^c=\varnothing$
• Union: $S\cup T=\{x|x\in Sorx\in T\}$
• Intersection: $S\cap T=\{x|x\in Sandx\in T\}$
• Disjoint: $S\cap T=\varnothing$

Some consequences of the definition

• $S\cup T=T\cup S$
• $S\cap (T\cup U)=(S\cap T)\cup (S\cap U)$
• $S\cup (T\cup U)=(S\cup T)\cup U$
• $S\cup (T\cap U)=(S\cup T)\cap (S\cup U)$
• $(S^c{)}^c=S$
• $S\cup \Omega =\Omega$
• $S\cap S^c=\varnothing$
• $S\cap \Omega =S$

DeMorgan Laws

$${\left(\bigcup _n{S}_n\right)}^c=\bigcap _n{S}_n^c$$

LHS: if $x\in {\left({\bigcup }_n{S}_n\right)}^c$ then it is not in any $S_n$ , thus for every $S_n$ it is in its complement.

$${\left(\bigcap _n{S}_n\right)}^c=\bigcup _n{S}_n^c$$

Probabilistic Model

Sample space

$\Omega$ - the set of all possible outcomes of an experiment

Probability law

A function $P()$ with input some $A\subset \Omega$ and output between 0 and 1 that encodes our knowledge or belief about the collective likelihood of the elements of $A$.

This subset $A$ is also called an event.

Notes

There is only one experiment, whether it’s 1 coin toss or 3.
Sample space can be finite or infinite (ex. discrete outcomes or continuous).
Outcomes/events are distinct/unique and mutually exclusive (cannot have an outcome be heads AND/OR tails).

Probability Axioms

1. Non-negativity: $P(A)\ge 0,\forall A$
2. Additivity: $P(A\cup B)=P(A)+P(B)$
   1. Assuming they are disjoint
3. Normalization: $P(\Omega )=1$

Uniform

$$P(A)=\frac{1}{N}$$

Where $\Omega$ has $N$ distinct events and $A$ is a distinct event.