Q: What does H2 Maths Notes (JC 1-2): 6) Probability and Statistics cover? A: Probability rules, discrete and normal distributions, sampling, hypothesis tests, 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, and regression features sit.
6.1 | Probability
Core ideas
A sample space lists all mutually exclusive outcomes; events are subsets of this space.
Addition rule: P(A∪B)=P(A)+P(B)−P(A∩B)
.
Conditional probability: P(A∣B)=P(B)P(A∩B) for any event B with P(B)>0.
Independent events satisfy P(A∩B)=P(A)P(B) and leave conditional probabilities unchanged.
Example -- Conditional reliability
A machine has sensors S1 and S2. The probability S1 triggers is 0.8, S2 is 0.7, and both triggering together is 0.6. Find:
P(S1∪S2)=0.8+0.7−0.6=0.9.
P(S1∣S2)=0.70.6≈0.857
Conclude the sensors are not independent because 0.8×0.7=0.56=0.6.
Exam habits
Sketch quick tree diagrams for sequential events; label branch probabilities cleanly.
When independence is in doubt, compute and compare P(A∩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)=∑xP(X=x) gives the long-run mean.
Variance Var(X)=E(X2)−[E(X)]2.
Binomial model: X∼Bin(n,p), where P(X=r)=(rn)pr(1−p)n−r
Example -- Binomial quality control
A factory tests n=10 items with defect probability p=0.08.
P(X=0)=(0.92)10≈0.434.
P(X≤2)≈0.930 using binomcdf(10, 0.08, 2).
Mean defects E(X)=np=0.8; variance np(1−p)=0.736.
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∼N(μ,σ2), standardise via Z=σX−μ∼N(0,1).
Symmetry: P(Z≤−a)=P(Z≥a).
Continuity correction bridges discrete-binomial to normal approximations using x±0.5.
Example -- Normal loading constraint
Shipping weights are modelled by X∼N(52,3.22). The crate fails if X>58.
Standardise: Z=3.258−52=1.875.
P(X>58)=1−Φ(1.875)≈0.030.
Interpret: roughly 3 percent of crates exceed the limit.
JC habits
Quote the standardisation line let Z = (X - 52)/3.2 before using distribution tables or GC.
When approximating binomial Bin(n,p), check np≥5 and n(1−p)≥5 before applying the normal model.
6.4 | Sampling and Estimation
Core ideas
A sample mean Xˉ from a large sample approximates N(μ,nσ2) by the Central Limit Theorem.
Unbiased estimator: E(Xˉ)=μ, E(S2)=σ2
Differentiate between sampling methods: simple random, stratified, systematic.
Example -- Estimating a mean
A JC samples n=40 students for weekly revision hours. Sample mean is 8.1 hours, sample standard deviation 1.4.
Standard error =401.4≈0.221.
A 95% confidence interval is 8.1±1.96×0.221=(7.67,8.53).
Interpret: the school is 95% confident the cohort mean lies in that interval.
6.5 | Hypothesis Testing
Framework
State H0 and H1 with parameter definitions.
Choose a test statistic (e.g. Z, t, or binomial count) and significance level α.
Compute the observed statistic and p-value or critical region.
Conclude by comparing to α and interpret in context.
Example -- One-tailed proportion test
A manufacturer claims the defect rate is p=0.08. A sample of n=50 finds x=7 defects. Test H0:p=0.08 against H1:p>0.08 at 5%.
Expectation np=4, variance np(1−p)=3.68. Normal approximation is acceptable; apply continuity correction.
Standardise: Z=3.686.5−4≈1.30.
p-value =1−Φ(1.30)≈0.097, which exceeds 0.05.
Do not reject H0; no evidence the defect rate increased.
Exam habits
Always write hypotheses with parameter notation, not in words alone.
Mention the distribution used explicitly (X ~ Bin(50, 0.08) then approx N(4, 3.68)).
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≤r≤1.
Regression line of y on x has equation y−yˉ=b(x−xˉ) where b=rSxSy.
Interpret r2 as the proportion of variance in y explained by the linear model.
Example -- Predicting exam scores
GC output gives r=0.82, sample means xˉ=65, yˉ=71, and gradient b=0.9 for Physics vs Maths marks.
Regression line: y−71=0.9(x−65)⇒y=0.9x+12.5.
Predict a student with x=70 scores y=0.9×70+12.5=75.5.
With r2=0.6724, about 67% of variation in Maths marks is explained by Physics marks.
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 execute continuity corrections.
Construct confidence intervals and justify sampling assumptions.
Perform hypothesis tests with full statements and contextual conclusions.
Interpret regression output, including gradient, intercept, r, and r2.
