# 1.1: Sample Spaces and Events

- Page ID
- 13542

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## Introduction

We begin with a definition.

### Definition \(\PageIndex{1}\)

* Probability theory* provides a mathematical model for chance (or random) phenomena.

While this is not a very informative definition, it does indicate the overall goal of this course, which is to develop a *formal, mathematical structure *for the fairly intuitive concept of probability. While most everyone is familiar with the notion of "chance" -- we informally talk about the chance of it raining tomorrow, or the chance of getting what you want for your birthday -- when it comes to *quantifying *the chance of something happening, we need to develop a mathematical model to make things precise and calculable.

## Sample Spaces and Events

Before we can formally define what the mathematical model is that we will use to make probability precise, we first establish the *structure* on which the model operates: *sample spaces* and *events*.

### Definition \(\PageIndex{2}\)

The * sample space* for a probability experiment (i.e., an experiment with random outcomes) is the set of all possible outcomes.

- The sample space is denoted \(\Omega\).
- An
is an*outcome**element*of \(\Omega\), generally denoted \(\omega \in \Omega\).

### Example \(\PageIndex{1}\)

Suppose we toss a coin twice and record the sequence of heads (\(h\)) and tails (\(t\)). A possible outcome of this experiment is then given by

$$\omega = ht\notag$$

and the sample space is

$$\Omega = \{hh, ht, th, tt\}.\label{coinflip}$$

### Example \(\PageIndex{2}\)

Suppose we record the time (\(t\)), in minutes, that a car spends waiting for a green light at a particular intersection. A possible outcome of this experiment is then given by

$$t=1.5,\notag$$

indicating that a particular car waited one and a half minutes for the light to turn green. The sample space consists of all non-negative numbers, since a measurement of time cannot be negative and, in theory, there is no limit on how a long a car could wait for a green light. We can then write the sample space as follows:

$$\Omega = \{t \in\mathbb{R}\ |\ t\geq 0\} = [0,\infty).\label{time}$$

### Definition \(\PageIndex{3}\)

An * event* is a particular subset of the sample space.

### Example \(\PageIndex{3}\)

Continuing in the context of Example 1.1.1, define \(A\)* *to be the event that at least one heads is recorded. We can write event \(A\) as the following subset of the sample space:

$$A = \{hh, ht, th\}.\notag$$

Note that \(A\) is a subset of \(\Omega\) given in Equation \ref{coinflip}.

### Example \(\PageIndex{4}\)

Continuing in the context of Example 1.1.2, define \(B\) to be the event that a car waits at most 2 minutes for the light to turn green. We can write the event \(B\) as the following interval, i.e., a subset of the sample space \(\Omega\) given in Equation \ref{time}:

$$B = [0,2] = \{t \in \mathbb{R}\ |\ 0 \leq t \leq 2\}.\notag$$

## Set Theory: A Brief Review

As we see from the above definitions of sample spaces and events, *sets* play the primary role in the structure of probability experiments. So, in this section, we review some of the basic definitions and notation from *set theory*. We do this in the context of sample spaces, outcomes, and events.

### Definition \(\PageIndex{4}\)

- The
of two events \(A\) and \(B\), denoted \(A\cup B\), is the set of all outcomes in \(A\) or \(B\) (**union***or both*). - The
of two events \(A\) and \(B\), denoted \(A\cap B\), is the set of all outcomes in both \(A\) and \(B\).**intersection** - The
of an event \(A\), denoted \(A^c\), is the set of all outcomes in the sample space that are not in \(A\). This may also be written as follows: $$A^c= \{\omega\in \Omega\ |\ s\notin A \}.\notag$$**complement** - The
**empty**, denoted \(\varnothing\), is the set containing no outcomes.**set** - Two events \(A\) and \(B\) are
(or**disjoint**) if their intersection is the empty set, i.e., \(A \cap B = \varnothing\).**mutually exclusive**

### Example \(\PageIndex{5}\)

Continuing in the context of both Examples 1.1.1 & 1.1.3, define \(B\) to be the event that exactly one heads is recorded:

$$B = \{ht, th\}.\notag$$

Now we can apply the set operations just defined to the events \(A\) and \(B\):

$$A \cup B = \{hh, ht, th\} = A\notag$$

$$A \cap B = \{ht, th\} = B\notag$$

$$A^c = \{tt\}\notag$$

$$B^c = \{hh, tt\}\notag$$

Note the relationship between events \(A\) and \(B\): every outcome in \(B\) is an outcome in \(A\). In this case, we say that \(B\) is a * subset* of \(A\), and write

$$B \subseteq A.\notag$$

Note also that events \(A\) and \(B\) are * *not * *disjoint, since their intersection is not the empty set. However, if we let \(C\) be the event that no heads are recorded, then

$$C = \{tt\},\notag$$

and

$$A \cap C = \varnothing\notag$$

$$B \cap C = \varnothing.\notag$$

Thus, events \(A\) and \(C\) are disjoint, and events \(B\) and \(C\) are disjoint.