Introduction
Probability is the language of uncertainty, and the 9709 syllabus demands that you can evaluate probabilities in two precise ways: by systematically listing all equally likely outcomes (enumeration), or by calculating the number of favourable and total outcomes using permutations and combinations. Both approaches rely on the same fundamental principle — so mastering when to use which method, and executing it accurately, is a high-value exam skill that underpins the entire Probability & Statistics unit.
Core Concept
The Classical (Equiprobable) Model
When every elementary event (single outcome) in a sample space is equally likely, the probability of an event is:
The sample space must be fully identified — either by enumeration (listing) or by counting with permutations/combinations.
Enumeration
For small sample spaces, list every possible outcome systematically. This guarantees completeness and avoids double-counting. Useful tools include:
- Lists (for one-stage experiments)
- Two-way tables (for two independent choices, e.g. rolling two dice)
- Tree diagrams (for sequential stages, especially with varying branches)
Counting with Permutations and Combinations
When the sample space is too large to list, count using:
| Situation | Formula | Use when… |
|---|---|---|
| Ordered arrangement of from | Order matters | |
| Unordered selection of from | Order does not matter |
The numerator counts favourable arrangements/selections; the denominator counts all possible ones.
Key judgement: If the problem involves "choosing a committee", "dealing cards", or "picking a group", order usually does not matter → use . If it involves "arranging letters", "finishing positions in a race", or "forming a code", order matters → use or with restrictions.
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