# 1.1: Sample Spaces and Events

- Page ID
- 3244

## 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 \(S\).
- An
is an*outcome**element*of \(S\), generally denoted \(s \in S\).

### 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

$$s = ht$$

and the sample space is

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

### 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

$$t=1.5,$$

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:

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

### 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:

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

Note that \(A\) is a subset of \(S\) 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 \(S\) given in Equation \ref{time}:

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

## 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: $$A^c= \{s\in S\ |\ s\notin A \}.$$**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:

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

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

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

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

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

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

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

$$B \subseteq A.$$

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

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

and

$$A \cap C = \varnothing$$

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

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