H2 Maths Probability & Statistics | Free Notes

Q: What does H2 Maths Statistics Notes: Probability and Statistics Overview cover?
A: Probability rules, binomial and normal models, sampling mean ideas, hypothesis tests for a population mean, and regression summarised for the 2026 H2 syllabus.

Before you dive in\ Reset your IP combinatorics and binomial knowledge so conditional probability and distribution work feel natural. Keep the GC set to Exam Mode and know where binompdf, binomcdf, normalcdf, invNorm, and regression features sit.

If you need one H2 Maths statistics page to revise before Paper 2, use this as the overview and then branch into the sub-topics below for targeted drills. For the full topic map, paper weightings, and official PDF link, see our H2 Maths Syllabus 2026-27 overview.

• Statistics turns data into decisions under uncertainty: Name the variable and context.
• Probability rules, distributions, tests, and regression each answer a different question: Identify the question type before using formulas.
• Paper 2 rewards interpretation as much as computation: End each answer with a plain-language statement.

Concrete example: "Find a probability" points to a distribution, while "is there evidence the mean changed?" points to hypothesis testing.

Status: SEAB's current H2 Mathematics (9758) syllabus PDF is labelled for 2026. Topic 6 is assessed in Paper 2 Section B (Probability and Statistics, 60 marks); Paper 1 is Pure Mathematics only.

Model-choice checkpoint

Before reaching for a calculator command, decide what kind of object the question is asking about. Most statistics errors start when a student uses the right formula family for the wrong question type.

Worked check: if a question says a factory defect count follows $X \sim \operatorname{Bin}(20,0.04)$, and asks for $P(X \leq 2)$, this is a discrete cumulative probability question. Start with binomcdf(20,0.04,2) or the corresponding binomial sum, not a normal curve or a hypothesis-test framework.

Misconception check: the graphing calculator does not choose the statistical model for you. Decide first whether the unknown is an event probability, a random variable, a population parameter, or a regression prediction. Then use the calculator to carry out the chosen model.

6.1 | Probability

Core ideas

• A sample space lists all mutually exclusive outcomes; events are subsets of this space.
• Addition rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
• Conditional probability: $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$

Example -- Conditional reliability

A machine has sensors $S_1$ and $S_2$. The probability $S_1$

1. $P(S_1 \cup S_2) = 0.8 + 0.7 - 0.6 = 0.9$

Exam habits

• Sketch quick tree diagrams for sequential events; label branch probabilities cleanly.
• When independence is in doubt, compute and compare $P(A \cap B)$ with $P(A)P(B)$.

6.2 | Discrete Random Variables

Definitions and notation

• A discrete random variable $X$ takes countable values with probability mass function $P(X = x)$.
• Expectation $E(X) = \sum x P\left( X = x \right)$

Example -- Binomial quality control

A factory tests $n = 10$ items with defect probability $p = 0.08$.

1. $P\left( X = 0 \right) = (0.92)^{10} \approx 0.434$.
2. $P(X \leq 2) \approx 0.960$

Calculator workflow

• Use binompdf(n, p, r) for exact terms, binomcdf(n, p, r) for cumulative sums.
• Record intermediate probabilities to three significant figures before rounding final answers.

6.3 | Normal Distribution

Essentials

• For $X \sim \mathcal{N}(\mu, \sigma^2)$, standardise via $Z = \frac{X - \mu}{\sigma} \sim \mathcal{N}(0, 1)$

Example -- Normal loading constraint

Shipping weights are modelled by $X \sim \mathcal{N}(52, 3.2^2)$. The crate fails if $X > 58$.

1. Standardise: $Z = \frac{58 - 52}{3.2} = 1.875$.
2. $P(X > 58) = 1 - \Phi(1.875) \approx 0.030$

JC habits

• Quote the standardisation line let Z = (X - 52)/3.2 before using distribution tables or GC.
• Normal approximation to binomial distribution is excluded in the 2026 syllabus, so do not introduce continuity correction in your solutions.

6.4 | Sampling and Estimation

Core ideas

• A sample mean $\bar{X}$ satisfies $E(\bar{X}) = \mu$ and $\operatorname{Var}(\bar{X}) = \sigma^2/n$

Example -- Sample mean probability

Assume weekly revision hours have $\mu = 8.1$ and $\sigma = 1.4$. For a random sample of $n = 40$, estimate $P(\bar{X} > 8.5)$

1. $\bar{X} \sim \mathcal{N}(8.1, 1.4^2/40) = \mathcal{N}(8.1, 0.049)$.
2. Standardise: $z = \frac{8.5 - 8.1}{\sqrt{0.049}} = 1.81$

6.5 | Hypothesis Testing

Framework

1. State $H_0$ and $H_1$ with parameter definitions.
2. Choose a test statistic $Z$ for a population mean and a significance level $\alpha$

Example -- One-tailed mean test

Historical mean attendance is 520 students. After a scheduling change, a sample of $n = 64$ lectures gives $\bar{X} = 508$, $\sigma = 40$. Test at 5% whether attendance decreased.

1. $H_0: \mu = 520$, $H_1: \mu < 520$

Exam habits

• Always write hypotheses with parameter notation, not in words alone.
• Mention the distribution or statistic used explicitly, such as $X \sim \operatorname{Bin}(50,0.08)$ for a binomial probability or $\bar{X}$ for a population-mean test.
• State the conclusion in everyday language linked to the context.

6.6 | Correlation and Linear Regression

Essentials

• Use the product-moment correlation coefficient $r$ to measure linear association; $-1 \leq r \leq 1$.
• Regression line of $y$ on $x$ has equation $y - \bar{y} = b(x - \bar{x})$

Example -- Predicting exam scores

GC output gives $r = 0.82$, sample means $\bar{x} = 65$, $\bar{y} = 71$

1. Regression line: $y - 71 = 0.9(x - 65) \Rightarrow y = 0.9x + 12.5$.
2. Predict a student with $x = 70$

Exam habits

• Quote the GC regression command (e.g. LinReg(ax+b) on TI) and state whether you are predicting $y$ from $x$ or vice versa.
• Avoid extrapolation beyond the observed $x$-range unless the question justifies it.

Practice Quiz

Assess your readiness across probability laws, distributions, inference, and regression before diving into the sub-topics.

6.7 | Quick Revision Checklist

• Manipulate conditional probabilities and independence tests confidently.
• Compute expectation and variance for discrete models, including binomial.
• Standardise normal variables and solve inverse-normal questions when needed.
• Use $E(\bar{X})$ and $\operatorname{Var}(\bar{X})$ (CLT) to model sample means.
• Perform hypothesis tests with full statements and contextual conclusions.
• Interpret regression output, including gradient, intercept, $r$

Sources

• SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 6 Probability and statistics (binomial model, normal distribution without binomial approximation, sample mean/CLT foundations, hypothesis tests for a population mean, correlation/regression without hypothesis tests). The official PDF link is recorded in the front matter citation for this note.

Next steps: Follow the H2 Maths notes hub into Sub-topic 6.1 - Probability and the rest of the statistics drill-down articles for full worked examples. If you want weekly Paper 2 support instead of self-study only, see our H2 Maths tuition Singapore page for timed drills, method-mark routines, and graphing-calculator practice.