Understanding Probability: Sample Space, Sample Points, and Events

Sample Space

The set of all possible outcomes of a random experiment is called the Sample Space, denoted by \( S \).

\( S \) is a set containing all possible outcomes of a random experiment.

Examples:

Tossing a single coin: \( S = \{H, T\} \), where \( H \) is “Head” and \( T \) is “Tail.”
Rolling a six-sided die: \( S = \{1, 2, 3, 4, 5, 6\} \), where each number represents the face value.
Tossing two coins: \( S = \{HH, HT, TH, TT\} \), where \( H \) and \( T \) represent outcomes of each coin.
Throwing a die and a coin together: \( S = \{H1, H2, H3, …, T6\} \), combining outcomes of both.

The size of the sample space, \( n(S) \), is the total number of possible outcomes. For example:

For a coin toss, \( n(S) = 2 \).
For rolling a die, \( n(S) = 6 \).
For tossing two coins, \( n(S) = 4 \).

Sample Point

Each individual outcome in the sample space is called a Sample Point.

Examples:

Tossing a coin: Sample points are \( H \) (Head) and \( T \) (Tail).
Rolling a die: Sample points are \( 1, 2, 3, 4, 5, 6 \), each representing one outcome.
Tossing two coins: Sample points are \( HH, HT, TH, TT \), representing all combinations of outcomes.
Throwing two dice: Sample points are ordered pairs, such as \( (1, 1), (1, 2), …, (6, 6) \).

The collection of all sample points forms the sample space. In complex experiments, each sample point corresponds to a unique combination of outcomes.

Event

An Event is a subset of the sample space \( S \), representing one or more possible outcomes.

An event is a collection of sample points from \( S \) that satisfies a certain condition.

Examples:

Tossing a coin:
- Event \( E \): Getting a head, \( E = \{H\} \).
- Event \( F \): Getting a tail, \( F = \{T\} \).
Rolling a die:
- Event \( E \): Getting an even number, \( E = \{2, 4, 6\} \).
- Event \( F \): Getting a number greater than 4, \( F = \{5, 6\} \).
Tossing two coins:
- Event \( E \): Getting at least one head, \( E = \{HH, HT, TH\} \).
- Event \( F \): Getting two tails, \( F = \{TT\} \).

Types of Events

Simple Event: Contains only one sample point. Example: Rolling a die and getting a 3, \( E = \{3\} \).
Compound Event: Contains multiple sample points. Example: Rolling a die and getting an odd number, \( E = \{1, 3, 5\} \).
Impossible Event: Contains no sample points. Example: Rolling a die and getting a 7, \( E = \{\} \) or \( E = \emptyset \).
Certain Event: Contains all sample points in \( S \). Example: Rolling a die and getting any number from 1 to 6, \( E = \{1, 2, 3, 4, 5, 6\} \).