Units 1 & 2 (Methods – Discrete Random Variables)
Focus on understanding discrete probability distributions, particularly for finite sample spaces. Emphasis is placed on probability rules, expected value, and real-world applications such as games of chance and decision making under uncertainty.

Unit 1: Foundations of Probability and Discrete Variables

1.1 Introduction to Discrete Random Variables

• Definition & Terminology: random variable, discrete vs. continuous, outcomes, events.

• Representing Distributions: probability tables, graphs (e.g. bar charts).

• Validity Conditions: probabilities must be non-negative and sum to 1.

1.2 Expected Value and Summary Statistics

• Calculating the Mean: E(X)=∑xP(x)E(X) = \sum xP(x)E(X)=∑xP(x)

• Interpreting Expected Value: long-run average, fair games

• Variance & Standard Deviation: understanding spread, using Var(X)=∑(x−E(X))2P(x)\text{Var}(X) = \sum (x - E(X))^2P(x)Var(X)=∑(x−E(X))2P(x)

1.3 Applications of Discrete Distributions

• Decision-Making Scenarios: comparing options using expected value

• Games of Chance: fair vs. biased games, risk vs. reward

• Use of Technology: spreadsheets or CAS tools to compute probabilities and statistics

1.4 Discrete Probability Rules

• Complementary Events: P(not A)=1−P(A)P(\text{not A}) = 1 - P(A)P(not A)=1−P(A)

• Addition Rule for Mutually Exclusive Events

• Basic Conditional Probability: when appropriate, via frequency tables or logical reasoning

Unit 2: Specific Distributions and Further Techniques

2.1 Binomial Distribution

• Conditions: fixed number of trials, two outcomes, constant probability, independent trials

• Calculating Probabilities

• Mean and Variance

2.2 Cumulative Probabilities and Expected Value

• Using Tables or Technology: cumulative binomial probability

• Applications: e.g. quality control, success/failure models

• Interpreting Results: context-based judgement

2.3 Problem Solving with Discrete Models

• Formulating Probability Models: translating from real contexts

• Evaluating Fairness and Strategy: cost-benefit analyses

• Modelling Assumptions: justifying use of discrete or binomial models

2.4 Simulation and Modelling Distributions

• Simulating Random Events: using random number generators

• Comparing Experimental vs. Theoretical Probabilities

• Application Tasks: using simulations to test fairness, estimate probability, or explore variation