H2 Maths Notes (JC 1-2): 6) Probability and Statistics

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: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \).
• Conditional probability: \( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \) for any event \(B\) with \( P(B) > 0 \).
• Independent events satisfy \( P(A \cap B) = P(A) P(B) \) and leave conditional probabilities unchanged.

Example -- Conditional reliability

A machine has sensors \(S_1\) and \(S_2\). The probability \(S_1\) triggers is \(0.8\), \(S_2\) is \(0.7\), and both triggering together is \(0.6\). Find:

1. \( P(S_1 \cup S_2) = 0.8 + 0.7 - 0.6 = 0.9 \).
2. \( P(S_1 \mid S_2) = \frac{0.6}{0.7} \approx 0.857 \).
3. Conclude the sensors are not independent because \( 0.8 \times 0.7 = 0.56 \neq 0.6 \).

Exam habits

• Sketch quick tree diagrams for sequential events; label branch probabilities cleanly.
• When independence is in doubt, compute and compare \( P(A \cap B) \) 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\( X = x \) \) gives the long-run mean.
• Variance \( \operatorname{Var}(X) = E(X^2) - [E(X)]^2 \).
• Binomial model: \( X \sim \operatorname{Bin}(n, p) \), where \( P\( X = r \) = \binom{n}{r} p^r (1 - p)^{n-r} \).

Example -- Binomial quality control

A factory tests \(n = 10\) items with defect probability \(p = 0.08\).

1. \( P\( X = 0 \) = (0.92)^{10} \approx 0.434 \).
2. \( P(X \leq 2) \approx 0.930 \) using binomcdf(10, 0.08, 2).
3. 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 \( X \sim \mathcal{N}(\mu, \sigma^2) \), standardise via \( Z = \frac{X - \mu}{\sigma} \sim \mathcal{N}(0, 1) \).
• Symmetry: \( P(Z \leq -a) = P(Z \geq a) \).
• Continuity correction bridges discrete-binomial to normal approximations using \( x \pm 0.5 \).

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. \( P(X > 58) = 1 - \Phi(1.875) \approx 0.030 \).
3. 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 \( \operatorname{Bin}(n, p) \), check \( np \geq 5 \) and \( n(1 - p) \geq 5 \) before applying the normal model.

6.4 | Sampling and Estimation

Core ideas

• A sample mean \( \bar{X} \) from a large sample approximates \( \mathcal{N}(\mu, \frac{\sigma^2}{n}) \) by the Central Limit Theorem.
• Unbiased estimator: \( E(\bar{X}) = \mu \), \( E(S^2) = \sigma^2 \) when \( S^2 \) is calculated with divisor \( n - 1 \).
• 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 \(1.4\).

1. Standard error \( = \frac{1.4}{\sqrt{40}} \approx 0.221 \).
2. A \(95%\) confidence interval is \( 8.1 \pm 1.96 \times 0.221 = (7.67, 8.53) \).
3. Interpret: the school is \(95%\) confident the cohort mean lies in that interval.

6.5 | Hypothesis Testing

Framework

1. State \( H_0 \) and \( H_1 \) with parameter definitions.
2. Choose a test statistic (e.g. \( Z \), \( t \), or binomial count) and significance level \( \alpha \).
3. Compute the observed statistic and p-value or critical region.
4. Conclude by comparing to \( \alpha \) 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 \( H_0: p = 0.08 \) against \( H_1: p > 0.08 \) at \(5%\).

1. Expectation \( np = 4 \), variance \( np(1 - p) = 3.68 \). Normal approximation is acceptable; apply continuity correction.
2. Standardise: \( Z = \frac{6.5 - 4}{\sqrt{3.68}} \approx 1.30 \).
3. p-value \( = 1 - \Phi(1.30) \approx 0.097 \), which exceeds \(0.05\).
4. Do not reject \( H_0 \); 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; \( -1 \leq r \leq 1 \).
• Regression line of \( y \) on \( x \) has equation \( y - \bar{y} = b(x - \bar{x}) \) where \( b = r \frac{S_y}{S_x} \).
• Interpret \( r^2 \) as the proportion of variance in \( y \) explained by the linear model.

Example -- Predicting exam scores

GC output gives \( r = 0.82 \), sample means \( \bar{x} = 65 \), \( \bar{y} = 71 \), and gradient \( b = 0.9 \) for Physics vs Maths marks.

1. Regression line: \( y - 71 = 0.9(x - 65) \Rightarrow y = 0.9x + 12.5 \).
2. Predict a student with \(x = 70\) scores \( y = 0.9 \times 70 + 12.5 = 75.5 \).
3. With \( r^2 = 0.6724 \), 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.

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 \( r^2 \).

Next steps: expand Topic 6 into sub-topic posts (6.1 through 6.6) with in-depth worked solutions, additional diagrams, and GC screenshots referenced in text.