H2 Maths Notes (JC 1-2): 00B) Section B -- Probability & Statistics Overview

> **How to use this guide**\\
> Section B supplies the statistical thinking that feeds subjects like Economics, Physics practicals, and university interviews. Use this roadmap to plan your sampling/hypothesis practice and to identify the exact MOE expectations before diving into the detailed notes.

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## 1 | Topic 6 Overview

The entire statistics strand is grouped under Topic 6. We break it into six focused articles (6.1-6.6) that mirror the MOE sub-topics.

| Sub-topic                     | Core competencies                                                                                     | Common pitfalls                                                                                 |
| ----------------------------- | ----------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------- |
| 6.1 Probability               | Counting arguments, conditional probability, independence proofs                                      | Mixing up mutually exclusive vs independent events; forgetting denominators in ( P(A \mid B) ). |
| 6.2 Discrete random variables | Probability mass functions, expectation/variance, binomial ( X \sim \operatorname\{Bin}(n, p) ) model | Using incorrect parameters, neglecting scenario fit (e.g. independence assumptions).            |
| 6.3 Normal distribution       | Standardisation, properties of ( \mathcal{N}(\mu, \sigma^2) ), linear combinations                    | Forgetting symmetry, mishandling variance scaling in ( \operatorname\{Var}(aX + bY) ).          |
| 6.4 Sampling                  | Sampling distributions, ( \bar{X} ), Central Limit Theorem                                            | Applying CLT without sufficient ( n ); mixing sample SD with population ( \sigma ).             |
| 6.5 Hypothesis testing        | Formulating ( H_0/H_1 ), critical regions, ( p )-values                                               | Swapping ( H_0 ) and ( H_1 ); misreporting conclusions in context.                              |
| 6.6 Correlation & regression  | PMCC, least squares, transformations                                                                  | Extrapolating beyond data range; interpreting correlation as causation.                         |

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## 2 | Sequencing Your Learning

1. **JC1 Term 1-2**: Counting techniques (permutations/combinations), baseline probability, introduction to discrete random variables.
2. **JC1 Term 3**: Binomial distribution applications, standard normal tables, first pass at sampling ideas.
3. **JC2 Term 1**: Hypothesis testing workflow and CLT-powered approximations; practise linking statistics to real-life contexts (e.g. chemistry titration data).
4. **JC2 Term 2**: Correlation/regression projects, statistical inference consolidation, exam drilling.

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## 3 | Modelling Toolkit

- **Probability algebra**: ( P(A \cup B) = P(A) + P(B) - P(A \cap B) ), ( P(A^c) = 1 - P(A) ), and the product rule for independent events.
- **Binomial mean/variance**: ( E(X) = np ), ( \operatorname\{Var}(X) = np(1 - p) ).
- **Normal standardisation**: ( Z = \frac\{X - \mu}\{\sigma} ) feeds the ( \mathcal{N}(0, 1) ) table; annotate diagrams to avoid sign mistakes.
- **Sampling variance**: ( \operatorname\{Var}(\bar{X}) = \frac\{\sigma^2}\{n} ); when ( \sigma ) is unknown, use the unbiased sample variance ( s^2 ) with divisor ( n - 1 ).
- **Regression line**: Use least squares to obtain \\( y = a + bx \\); check residuals before trusting predictions.

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## 4 | Examination Strategy

- **Structure**: Each paper features statistics questions that demand interpretation, not just number crunching. Quote conclusions in plain English referencing the context (e.g. "There is sufficient evidence at the 5% level that the mean tensile strength exceeds ( 4.2 \pu\{kN} ).").
- **Calculator readiness**: Practise binomial/normal/PMCC commands on the GC; store sequences for quick re-computation.
- **Worked example workflow**
  1. Sketch: tree diagram, distribution, or scatter plot.
  2. Compute with precise notation (e.g. ( P\\( X = 3 \\) = \binom\{10}\{3} 0.3^3 0.7^7 )).
  3. Conclude with units and wording tied to the scenario.

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## 5 | What to Review from IP/O-Level

- Additional Maths probability identities.
- Sampling language from O-Level statistics (mean, variance formulations).
- Linear graphs/transformations (to accelerate regression understanding).

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## 6 | Next Steps

Dive into the detailed Topic 6 posts; each will provide:

- Digestible concept summaries aligned with MOE's bullet list.
- Step-by-step worked examples with calculator screenshots (where relevant).
- Exam-style practice with annotated solutions.

Start with `H2 Maths Notes (JC 1-2): 6.1) Probability` once it is published.