## Unit 1

In this area of study, students cover the concepts of event, frequency, probability and representation of finite sample spaces and events using various forms such as lists, grids, Venn diagrams, Karnaugh maps, tables and tree diagrams. This includes consideration of impossible, certain, complementary, mutually exclusive, conditional and independent events involving one, two or three events (as applicable), including rules for computation of probabilities for compound events.

This area of study includes:

- random experiments, sample spaces, outcomes, elementary and compound events
- simulation using simple random generators such as coins, dice, spinners and pseudo-random generators using technology, and the display and interpretation of results, including informal consideration of proportions in samples
- probability of elementary and compound events and their representation as lists, grids, Venn diagrams, Karnaugh maps, tables and tree diagrams
- the addition rule for probabilities, \( \Pr(A \cup B) = \Pr(A) + \Pr(B)–\Pr(A \cap B) \), and the relation that for mutually exclusive events \( \Pr(A \cap B) = 0 \), hence \( \Pr(A \cup B) = \Pr(A) + \Pr(B) \)
- conditional probability in terms of reduced sample space, the relations \( \Pr(A | B) = \displaystyle \frac{\Pr(A \cup B)}{\Pr(B)} \) and \( \Pr(A \cap B) = \Pr(A | B) \times \Pr(B) \)
- the law of total probability for two events \( \Pr(A) = \Pr(A | B) \Pr(B) + \Pr(A | B^{\prime}) \Pr(B^{\prime}) \)
- the relations that for pairwise independent events \( A \) and \( B \), \( \Pr(A | B) = \Pr(A), \Pr(B | A) = \Pr(B) \) and \( \Pr(A \cap B) = \Pr(A) \times \Pr (B)\) .

## Unit 2

In this area of study, students cover introductory counting principles and techniques and their application to probability and the law of total probability in the case of two events.

This area of study includes:

- addition and multiplication principles for counting
- combinations: concept of a selection and computation of \(^n C_r \) application of counting techniques to probability.

*source – VCE Mathematics Study Design*

#### Introduction to Probability

- Video - Basic Probability (11:40)
- Video - Standard Pack of 52 Cards (11:18)
- Topic - Basic Properties of Probability
- Topic - Informal Description of Chance: Drag
- Topic - Informal Description of Chance: Multiple Choice
- Topic - Informal Description of Chance: True or False
- Topic - Informal Description of Chance: Likely

#### Experimental Probability

- Video - Estimating Probabilities from Data (5:13)
- Video - Understanding Regular Pack of 52 Cards (2:52)
- Video - Solving Regular Pack of 52 Cards Problems involving Probability (4:57)
- Video - Probability of Choosing All Different Suits from Regular Pack of 52 Cards (3:31)
- Video - Probability involving Dice (7:22)
- Video - Probability involving Sum of Faces of Dice (2:20)
- Video - Probability involving Product of Faces of Dice (6:01)
- Topic - Counting Number of Cards
- Topic - Probability using Regular Pack of 52 Cards
- Topic - Two Digit Numbers by Dice
- Topic - Sum of Two Dice

#### Sampling With Replacement

- Video - Probability with Replacement (6:07)
- Video - Probability with Replacement Excluding Duplicates (3:33)
- Video - Probability of Events with Replacements 1 (2:36)
- Video - Probability of Events with Replacements 2 (2:14)
- Video - Probability of Events with Replacements 3 (1:48)
- Topic - Sampling with Replacement

#### Complementary Events

- Video - Understanding Complementary Events (15:28)
- Video - Probability of Events using At Least 1 (4:31)
- Video - Probability of Events using At Least 2 (8:49)
- Video - Probability of Events using At Most (2:00)
- Video - Arithmetic Calculation of Probability (1:22)
- Topic - Probability with At Least
- Topic - Probability with Ratios
- Topic - Probability of Complement Events without Replacement

#### Sampling Without Replacement

- Video - Probability Without Replacement (5:57)
- Video - Probability without Replacement involving Ordering Events (4:30)
- Video - Probability Without Replacement involving Counting Techniques (4:55)
- Video - Probability without Replacement involving Specific Orders (6:42)
- Video - Probability without Replacement involving Selection of Items (6:04)
- Video - Probability without Replacement involving Raffles (4:01)
- Video - Probability of Events without Replacements (4:07)
- Video - Representing Events Without Replacements in Venn Diagrams (4:48)
- Video - Tree Diagrams involving Without Replacement (5:49)
- Topic - Sampling without Replacement
- Topic - Probability without Replacement with Complement Events
- Topic - Probability without Replacement with AND OR
- Topic - Probability without Replacement and Order of Events

#### Laws of Probability

- Video - Addition of Probabilities (9:59)
- Video - Addition Rule of Probability (3:59)
- Video - Addition Rule of Probability involving Mutually Exclusive Events (5:53)
- Video - Product Rule of Probability (5:28)
- Video - Product Rule of Probability Involving Repeated Occurrences (5:34)
- Video - Probabilities involving Addition Rule and Product Rule (2:18)
- Video - Probability involving Number Digits 1 (5:40)
- Video - Probability involving Number Digits 2 (1:28)
- Topic - Addition Law of Probability
- Topic - Probability Regarding OR
- Topic - Addition Rule of Probability and Mutually Exclusive Events
- Topic - Probability Regarding AND
- Topic - Product Rule and Independent Events

#### Conditional Probability

- Video - Theory of Conditional Probability (5:46)
- Video - Techniques of Conditional Probability (1:30)
- Video - Conditional Probability in Multiple Events (16:32)
- Video - Conditional Probability in Continuous Random Variables (4:35)
- Video - Conditional Probability in Venn Diagrams (4:46)
- Topic - Conditional Probability Formula 1
- Topic - Conditional Probability Formula 2

#### Dependent Events

- Video - Dependent Events involving Orders (1:47)
- Video - Dependent Events in Choosing Books (5:42)
- Video - Dependent Events involving Birthdays (2:00)
- Video - Probability involving Dependent Events 1 (2:38)
- Video - Probability involving Dependent Events 2 (3:12)
- Video - Probability involving Dependent Events 3 (3:27)

#### Independent Events

- Video - Definition of Independent Events (7:09)
- Video - Basic Independent Events (4:46)
- Video - Independent Events in Tables (2:05)
- Video - Independent Events in Ordering Events (4:44)
- Video - Independent and Compound Events (4:07)
- Video - A Pair of Regular Dice (3:46)
- Video - Independent Events (1:00)
- Video - Probability involving Independent Events 1 (1:37)
- Video - Probability involving Independent Events 2 (3:10)
- Video - Probability involving Independent Events 3 (3:54)
- Video - Probability involving Independent Events 4 (1:33)
- Topic - Determining Independent & Dependent Events
- Topic - Pairing Independent Events

#### Tree Diagrams

- Video - Basic Tree Diagrams (5:03)
- Video - Tree Diagrams involving Percentages (7:45)
- Video - Tree Diagrams involving Independent Events (3:44)
- Video - Tree Diagrams involving Compound Events (7:15)
- Video - Tree Diagrams involving Twice as Likely (6:31)
- Video - Asymmetrical Tree Diagrams (2:10)
- Video - Probability involving Probability Tree 1 (2:34)
- Video - Probability involving Probability Tree 2 (2:35)
- Video - Probability involving Probability Tree 3 (2:39)
- Topic - Play with Tree Diagrams

#### Sets and Venn Diagrams

- Video - Understanding Venn Diagrams (2:43)
- Video - Drawing Venn Diagrams and Sets (2:20)
- Video - Drawing Venn Diagrams and Subsets (0:55)
- Video - Drawing Venn Diagrams and Mutually Exclusive Sets (0:48)
- Video - Representing Data in Venn Diagrams (3:40)
- Video - Probabilities in Venn Diagrams (5:01)
- Topic - Sets and Venn Diagrams

#### Product Principle

- Video - Arrangements with Repetitions: Numeric Passwords (4:53)
- Video - Arrangements with Repetitions: Fixed Spot Alphanumeric Passwords (2:07)
- Video - Arrangements with Repetitions: Floating Spot Alphanumeric Passwords (3:10)
- Video - Arrangements with Repetitions: Combination Locks (1:35)
- Video - Arrangements with Repetitions: Lights On/Off (1:54)
- Video - Arrangements with Repetitions: Binary Numbers (1:46)
- Video - Arrangements without Repetitions: Numeric Passwords (3:06)
- Video - Arrangements without Repetitions: Alphanumeric Passwords (2:14)
- Video - Arrangements without Repetitions: Combination Locks (1:25)

#### Factorials

- Video - Definition of Factorial (5:52)
- Video - Simplifying Expressions involving Factorials (4:22)
- Video - Equations involving Factorials (2:08)
- Video - Arrangements of 4 Cards (2:04)
- Video - Arrangements of 9 People Queuing Up a Line (5:18)
- Video - Arrangements of 6 Girls and 2 Boys (4:09)
- Video - Formula of Considering Duplicates (2:57)
- Video - Arrangements involving Duplicates (3:01)
- Video - Arrangements involving Duplicates in Groups (1:11)
- Topic - Calculating Factorials
- Topic - Factorials involving Fractions
- Topic - Factorials involving Multi Factorials & Fractions

#### Combinations

- Video - Counting Techniques without Replacements (2:28)
- Video - Unordered Selections: Choosing Letters (8:21)
- Video - Unordered Selections: Forming Groups (3:03)
- Video - Definition of Combination (3:59)
- Video - Simplifying Combinations (3:25)
- Video - Equations involving Combination (3:46)
- Video - Proofs involving Combination (3:30)
- Video - Understanding Combination (1:49)
- Video - Choosing Combinations from Two Groups (4:17)
- Video - Combinations involving At Least (2:26)
- Video - Combinations involving Duplicates (5:05)