Probability and Events Revision Notes for HSC SSCE Mathematics Advanced

Probability and Events

Introduction

When working with probability, we need to understand the fundamental concepts of experiments, outcomes, and events. This topic introduces the mathematical framework for calculating probability when all outcomes are equally likely.

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The methods in these notes apply specifically to situations where each outcome has an equal chance of occurring. This is a fundamental assumption in basic probability theory.

Experiments and sample spaces

What is a random experiment?

A random experiment is a process that has an observable result. The key feature is that we can repeat the experiment under the same conditions, even though we cannot predict the exact outcome in advance. Examples include rolling a die, tossing a coin, or drawing a card from a deck.

Outcomes and sample spaces

An outcome is any possible result that can occur from performing an experiment or trial.

The sample space, denoted by $S$ , is the complete set of all possible outcomes from a random experiment. We write this using set notation. For example, when flipping a coin, the sample space is $S = \{\text{heads, tails}\}$ . When tossing two identical coins simultaneously, the sample space is $S = \{HH, HT, TT\}$ , where $H$ represents heads and $T$ represents tails.

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The cardinality of the sample space, written as $|S|$ , tells us the total number of possible outcomes. Think of cardinality as simply "counting" how many elements are in the set.

Trials

A trial is a single performance of a random experiment. When we conduct successive trials, we repeat the same experiment multiple times. Each trial has the same sample space because the possible outcomes remain constant.

Events

An event is a subset of the sample space. We can write this relationship as $A \subseteq S$ , meaning event $A$ is contained within sample space $S$ .

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If an event contains no outcomes, it is represented by the empty set $\varnothing$ . This represents an impossible event - something that cannot happen given the conditions of the experiment.

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Worked Example: Card Deck Sample Space

Let's consider an experiment where we draw a card from a standard deck of 52 cards.

Finding the cardinality:

The sample space $S$ consists of all 52 cards in the deck.

$S = \{\text{all 52 cards}\}$

Therefore: $|S| = 52$

Finding the size of an event:

Let $A$ be the event that a heart is drawn.

Since there are 13 hearts in a standard deck:

$A = \{\text{13 hearts}\}$

Therefore: $|A| = 13$

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Key Concepts Summary

An experiment or trial is a repeatable procedure with a sample space $S$ , which represents the set of all possible outcomes.

An event is a subset of $S$ , where $|S|$ gives the number of outcomes. The empty set $\varnothing$ represents an impossible event.

Probability of events

Understanding probability

Probability measures the chance of something happening, expressed on a scale from 0 to 1 (inclusive). A probability of 0 means the event is impossible, while a probability of 1 means the event is certain. For example, the probability of getting heads when tossing a fair coin is 0.5.

The probability formula

When all outcomes in a sample space $S$ are equally likely, we can calculate the probability of an event $A$ using the ratio of favorable outcomes to total outcomes:

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The Fundamental Probability Formula:

$P(A) = \frac{|A|}{|S|}$

where:

$P(A)$ is the probability of event $A$ occurring
$|A|$ is the number of outcomes in event $A$
$|S|$ is the number of outcomes in the sample space $S$

This formula only applies when all outcomes are equally likely!

Set notation in probability

We use specific notation to describe different types of events. Understanding these symbols is essential for working with probability:

$\bar{A}$ represents the complement of $A$ , meaning " $A$ does not occur"
$A \cap B$ represents the intersection of $A$ and $B$ , meaning "both $A$ and $B$ occur"
$A \cup B$ represents the union of $A$ and $B$ , meaning " $A$ or $B$ occurs" (or both)

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Memory Aid for Set Notation:

Complement ( $\bar{A}$ ) = Complete opposite = NOT $A$
Intersection ( $A \cap B$ ) = In both sets = $A$ AND $B$
Union ( $A \cup B$ ) = U-nited together = $A$ OR $B$

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Worked Example: Drawing Balls from a Bag

A bag contains 4 red balls and 6 blue balls. A ball is drawn at random. Let $A$ be the event of drawing a red ball and $B$ be the event of drawing a blue ball.

Part (a): Finding the probability of drawing a red ball

First, identify the sample space:

$|S| = 4 + 6 = 10$

Event $A$ (red ball) contains 4 outcomes:

$|A| = 4$

Apply the probability formula:

$P(A) = \frac{|A|}{|S|} = \frac{4}{10} = \frac{2}{5}$

The probability of drawing a red ball is $\frac{2}{5}$ .

Part (b): Finding the complement probability

We need to find $P(\bar{A})$ , which represents drawing a ball that is not red (i.e., a blue ball).

Use the complement formula:

$P(\bar{A}) = 1 - P(A) = 1 - \frac{2}{5} = \frac{3}{5}$

The probability of drawing a blue ball is $\frac{3}{5}$ .

Part (c): Finding the union probability

We need to find $P(A \cup B)$ , which represents drawing either a red or blue ball.

Since only red or blue balls exist in the bag:

$A \cup B = S$

Apply the probability formula:

$P(A \cup B) = \frac{|A \cup B|}{|S|} = \frac{|S|}{|S|} = 1$

This makes sense because every ball drawn must be either red or blue, so the probability is 1 (certain).

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Worked Example: Rolling a Die

A die is rolled. Let $A$ be the event of rolling an odd number, and $B$ be the event of rolling a number less than 4.

Part (a): Finding the intersection probability

First, identify the sample space:

$S = \{1, 2, 3, 4, 5, 6\}$

Therefore: $|S| = 6$

Define event $A$ (odd numbers):

$A = \{1, 3, 5\}$

Therefore: $|A| = 3$

Define event $B$ (numbers less than 4):

$B = \{1, 2, 3\}$

Therefore: $|B| = 3$

Find the intersection (numbers that are both odd AND less than 4):

$A \cap B = \{1, 3\}$

Therefore: $|A \cap B| = 2$

Calculate the probability:

$P(A \cap B) = \frac{|A \cap B|}{|S|} = \frac{2}{6} = \frac{1}{3}$

Part (b): Finding the union probability

Combine the elements from both sets:

$A \cup B = \{1, 2, 3, 5\}$

Therefore: $|A \cup B| = 4$

Calculate the probability:

$P(A \cup B) = \frac{|A \cup B|}{|S|} = \frac{4}{6} = \frac{2}{3}$

Checking the answer:

We can verify this result using the inclusion-exclusion principle:

|A \cup B| = |A| + |B| - |A \cap B| = 3 + 3 - 2 = 4 \checkmark

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Key Concepts Summary

The probability of an event $A$ is calculated as $P(A) = \frac{|A|}{|S|}$ when outcomes are equally likely.

Set notation helps us describe events precisely:

$\bar{A}$ means " $A$ does not occur"
$A \cap B$ means " $A$ and $B$ occur"
$A \cup B$ means " $A$ or $B$ occurs"

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Remember! Key Points to Remember:

The sample space $S$ contains all possible outcomes of an experiment, and $|S|$ tells us how many outcomes there are
An event is any subset of the sample space, representing a particular outcome or group of outcomes we're interested in
The probability formula $P(A) = \frac{|A|}{|S|}$ works when all outcomes are equally likely
The complement $\bar{A}$ represents the event not happening, and $P(\bar{A}) = 1 - P(A)$
Set notation is essential: use $\cap$ for "and" (intersection), $\cup$ for "or" (union), and $\bar{A}$ for "not" (complement)

Probability and Events (HSC SSCE Mathematics Advanced): Revision Notes