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01. Introduction

$\gdef \N{\mathbb{N}} \gdef \Z{\mathbb{Z}} \gdef \Q{\mathbb{Q}} \gdef \R{\mathbb{R}} \gdef \C{\mathbb{C}} \gdef \setcomp#1{\overline{#1}} \gdef \sseq{\subseteq} \gdef \pset#1{\mathcal{P}(#1)} \gdef \covariant{\operatorname{Cov}} \gdef \of{\circ} \gdef \p{^{\prime}} \gdef \pp{^{\prime\prime}} \gdef \ppp{^{\prime\prime\prime}} \gdef \pn#1{^{\prime\times{#1}}} $

General Concepts

  • What is probability?
    • It’s a measure of how likely something is to happen
Definitions
Experiments

An experiment is some occurance which can turn out in multiple ways

  • Rolling dice
  • Flipping coins
  • Anything in a random manner
  • Weather
Outcomes

An outcome is a discrete result of a given experiment.

Sample Spaces

The set of all possible outcomes is called the sample space, denoted as \(\Omega\)

  • Eg., Rolling a die, \(\Omega = \{1,2,3,4,5,6\}\)
  • Eg., Flipping a coin, \(\Omega=\{H,T\}\)
  • Rolling 2 dice — more complicated, it depends what you want.
    • Sum of rolls: \(\Omega=\{2,3,4,\dots,12\}\)
    • Distinct results: \(\Omega=\{(1,1),(1,2)\dots(1,6)\dots(6,6)\}\)
Events

An event is a subset of \(\Omega\)

  • It’s a set of possible outcomes.
    • Eg, rolling a single die < 6
      • \(S=\{1,2,3,4,5\}\)
  • Other events:
    • \(\Omega\)
    • \(\varnothing\)
    • \(\{a\}\mid a\in\Omega\) (single outcomes)
  • A single outcome is a simple/elementary event
  • The size of an event \(A\) is denoted as \(|A|\)
  • Disjoint events are events that cannot occur at the same time.
    • Events \(A\) and \(B\) are said to be disjoint iff \(A\cap B=\varnothing\)
    • A coin landing heads up is disjoint with a coin landing tails up.
Probability Functions

A probability function assigns a number \(\in[0,1]\) to every event in \(\Omega\).
\(P(A)\) (the probability of event A) measures how likely the outcome is to be an element of \(A\).

  • In all cases, \(P(\Omega)=1\), and \(P(\varnothing)=0\)
Uniform Sample Spaces

A uniform sample space (USS) is a sample space where each outcome is equally likely.
In a (finite) USS \(S\):

\[P(S)=\frac{|S|}{|\Omega|}\]

Evaluating Probabilities in a Uniform Sample Space

2 dice are rolled.
Let \(S\) be the event: Rolling \(< 6\).
What is \(P(S)\)?

Using the “distinct results” definition of \(\Omega\), we can calculate:

\[P(S)=\frac{|S|}{|\Omega|}=\frac{\text{number of outcomes in }S}{\text{number of outcomes in }\Omega}=\frac{10}{36}=\frac{5}{18}\]
Complementary Events

Two events are said to be complementary when one event occurs if and only if the other does not. The probabilities of two complimentary events add up to 1.

If \(S_1,S_2\) are complimentary events, then:

  • \(|S_1|+|S_2|=|\Omega|\)
  • \(P(S_1\cup S_2)=1\)
  • \(P(S_1)+P(S_2)=1\)