Sample Spaces and Probability (VCE SSCE Mathematical Methods): Revision Notes

Sample Spaces and Probability

Introduction

Probability is built on two essential components: the sample space, which represents all possible outcomes of an experiment, and a set of probabilities, one assigned to each outcome. Understanding these concepts forms the foundation for working with probability.

Sample spaces

When we perform an experiment with uncertain outcomes, we call this a random experiment. A single performance of this experiment is known as a trial. Although we cannot predict the exact outcome before conducting the trial, we know the outcome must be one from a specific set of possibilities.

The sample space is the complete set of all possible outcomes that could occur in a random experiment. We use the Greek letter $\varepsilon$ (epsilon) to denote the sample space throughout probability.

Notation and representation

We express sample spaces using set notation with curly brackets. For example, when tossing a coin, the sample space is:

$$\varepsilon = \{H, T\}$$

where $H$ represents heads and $T$ represents tails.

Types of sample spaces

Sample spaces can be classified in two ways:

Finite or infinite:

• A finite sample space contains a countable, limited number of outcomes
• An infinite sample space contains unlimited outcomes

Discrete or continuous:

• A discrete sample space has separate, distinct outcomes (like counting numbers)
• A continuous sample space has outcomes forming a continuous range (like time or distance)

Examples of sample spaces

The following table shows various random experiments and their corresponding sample spaces:

Random experiment	Sample space
The number observed when a die is rolled	$\varepsilon = \{1, 2, 3, 4, 5, 6\}$
The number of brown eggs in a carton of 12 eggs	$\varepsilon = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$
The result when two coins are tossed	$\varepsilon = \{HH, HT, TH, TT\}$
The number of calls to your phone in the next two hours	$\varepsilon = \{0, 1, 2, 3, 4, \ldots\}$
The time, in hours, it takes to complete your homework	$\varepsilon = \{t : t \geq 0\}$

The first three examples show discrete, finite sample spaces. The phone calls example is discrete but infinite (indicated by the dots). The homework time example is continuous (time can take any non-negative value).

Events

An event is a subset of the sample space. It represents a particular outcome or collection of outcomes that we are interested in. Events can consist of a single outcome or multiple outcomes.

We use capital letters ($A$, $B$, $C$, etc.) to denote events and express them using set notation.

Examples of events

Consider rolling a die where $\varepsilon = \{1, 2, 3, 4, 5, 6\}$:

• The event "getting a six" = $\{6\}$ (single outcome)
• The event "getting an odd number" = $\{1, 3, 5\}$ (multiple outcomes)

The table below shows sample spaces with corresponding events:

Sample space	An event
Rolling a die: $\varepsilon = \{1, 2, 3, 4, 5, 6\}$	"An even number" = $\{2, 4, 6\}$
Brown eggs in a carton: $\varepsilon = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$	"More than half brown" = $\{7, 8, 9, 10, 11, 12\}$
Two coins tossed: $\varepsilon = \{HH, HT, TH, TT\}$	"Two heads" = $\{HH\}$
Phone calls in two hours: $\varepsilon = \{0, 1, 2, 3, 4, \ldots\}$	"Fewer than two phone calls" = $\{0, 1\}$
Homework time: $\varepsilon = \{t : t \geq 0\}$	"More than two hours" = $\{t : t > 2\}$

Determining probabilities for equally likely outcomes

In many situations, we can reasonably assume all outcomes are equally likely to occur. For example, when rolling a fair die, each number from 1 to 6 has the same chance of appearing.

Probability for equally likely outcomes

When the sample space $\varepsilon$ contains $n$ outcomes that are all equally likely, we assign a probability of $\frac{1}{n}$ to each outcome.

For example, when rolling a die with six equally likely outcomes:

$$\text{Pr}(1) = \text{Pr}(2) = \text{Pr}(3) = \text{Pr}(4) = \text{Pr}(5) = \text{Pr}(6) = \frac{1}{6}$$

Calculating event probabilities

The probability of an event $A$ equals the sum of the probabilities of all outcomes in that event.

For equally likely outcomes, we can use the formula:

$$\text{Pr}(A) = \frac{n(A)}{n(\varepsilon)} = \frac{m}{n}$$

where:

• $n(A)$ represents the number of elements in event $A$ (= $m$)
• $n(\varepsilon)$ represents the total number of elements in the sample space (= $n$)

Example: Let $A$ be the event that an even number is rolled on a die.

$A = \{2, 4, 6\}$, so $n(A) = 3$ and $n(\varepsilon) = 6$

$$\text{Pr}(A) = \frac{3}{6} = \frac{1}{2}$$

Rules of probability

For finite sample spaces, probability must follow two fundamental rules:

These rules help us determine unknown probabilities.

Probabilities with spinners

When working with spinners divided into sectors, we can determine probabilities based on the relative sizes of the sectors.

Complementary events

Complementary events are two events that have no elements in common and together make up the entire sample space. The complement of event $A$ is written as $A'$ (read as "A prime" or "A complement").

The complement $A'$ contains all outcomes in $\varepsilon$ that are not in $A$.