- 1. What Is Probability? — Language, Scale, and Outcomes
Define and use the language of probability — experiment, trial, outcome, sample space (S, n(S)), and event (E, n(E)) — and place and classify events on the 0-to-1 probability scale as impossible, unlikely, equally likely, likely, or certain.
- 2. Calculating Probability — The Formula P(E) = n(E)/n(S)
State and apply the theoretical-probability formula P(E) = n(E)/n(S) using a five-step procedure, and find probabilities for dice, integer-range, letter/word, and bag experiments — including the inclusive-range count n(S) = upper − lower + 1.
- 3. Two Dice and the Grid — Building Compound Sample Spaces
Use the Multiplication Principle to find n(S) for combined experiments, construct and read a systematic 6 × 6 outcome grid for two dice, and calculate probabilities of events defined by sums, equal faces, comparisons, and 'at least one' conditions.
- 4. Mapping Every Outcome — Tree Diagrams and Coin Tosses
Construct tree diagrams for one-, two-, and three-stage coin experiments, use the 2^n pattern to find n(S) for any number of tosses, and calculate probabilities for 'exactly k', 'at least k', and 'at most k' events directly from the tree.
- 5. The Shortcut That Always Works — The Complement Rule
Define and identify the complement E' of any event, state and apply the Complement Rule P(E') = 1 - P(E), and recognise 'not', 'neither...nor', and 'at least one' problems as complement problems and solve them efficiently.
- 6. Probability Across Contexts — Balls, People, and Playing Cards
Describe the structure of a standard 52-card deck and use it as a sample space, apply P(E) = n(E)/n(S) and the Complement Rule to single-draw card events, and calculate probabilities by suit, colour, value, and combined conditions across cards, balls, and people.
- 7. Mutually Exclusive Events and the Addition Rule
Define mutually exclusive events and identify them using P(A ∩ B) = 0, apply the Addition Rule P(A ∪ B) = P(A) + P(B) for mutually exclusive events, and recognise the 'or' / 'either...or' signal.
- 8. When Events Overlap — The General Addition Rule
Identify overlapping (non-mutually-exclusive) events where A ∩ B ≠ ∅, state and apply the General Addition Rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B), and use it in both directions to find P(A ∪ B) or P(A ∩ B).
- 9. Independent Events — The Multiplication Rule and With Replacement
Define independent events and state P(A ∩ B) = P(A) × P(B), recognise 'and / both' language and apply the multiplication rule (including three or more events), and explain why drawing 'with replacement' guarantees independence.
- 10. Without Replacement — Sequential Dependent Events
Define dependent events and explain why drawing without replacement creates dependence, apply P(A then B) = P(A) × P(B from the updated sample space), and solve 'same colour', 'different colour', and reverse-engineering problems using the complete decision framework.
